Riesz Projection and Essential $ S $-Spectrum in Quaternionic Setting

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2006, F. Colombo and I. Sabadini succeeded in devising a new notion con- ducive to the complete development of the quaternionic operator theory, namely the S-spectrum. We refer to [13, Subsection 1.2.1] for a precise history. After several more years of formulating the spectral theorem that stemmed from the S-spectrum, D. Alpay, F. Colombo, D.P. Kimsey, see [6], supplied ample evidence in 2016 to further establish this fundamental theorem for both bounded and unbounded operators. In the book [13], see also [14], the authors briefly explain the concept of S−spectrum and give the systematic basis of quaternionic spectral theory. We refer to the book [3] and the references therein for the spectral theory on the S-spectrum for Clifford operators and to [9] for some results on operators perturbation. Motivated by the new concept of S−spectrum in the quaternionic setting, Muraleetharan and Thiralogasanthar, in [29, 30], introduced the Weyl and essential S-spectra and gave a characterization using Fredholm operators. We refer to [7] for the study of the general framework of the Fredholm element with respect to a quaternionic Banach algebra homomorphism. In general, the set of all operators acting on right Banach space is not quaternionic Banach algebra with respect to the composition operators. By [19, R Theoreme 7.1 and Theorem 7.3], if VH is a separable quaternionic Hilbert R space, then B(VH ) (the set of all right bounded operators) is a quaternionic ∗ R two-sided C −algebra and the set of all compact operators K(VH ) is a closed R two-sided ideal of B(VH ) which is closed under adjunction. In this regard, in [29], the author defined the essential S−spectrum as the S−spectrum of quotient map image of bounded right linear operator on the Calkin algebra R R B(VH )/K(VH ). In order to explain the objective of this work, we start by recalling a few results concerning the discrete spectrum and the Riesz projection in the complex setting. Let T be a linear operator acting on a complex Banach space VC. We denote the spectrum of T by σ(T ). Let σ be an isolated part of σ(T ). The Riesz projection of T corresponding to σ is the operator

1 −1 Pσ = (z − T ) dz 2πi ZCσ where Cσ is a smooth closed curves belonging to the resolvent set C\σ(T ) such that Cσ surrounds σ and separates σ from σ(T )\σ. The discrete spectrum of T , denoted σd(T ), is the set of isolated point λ ∈ C of σ(T ) such that the corresponding Riesz projection P{λ} are ﬁnite dimensional, see [21, 28]. Note that in general we have σe(T ) ⊂ σ(T )\σd(T ), where σe(T ) denotes the set of essential spectrum of T . We refer to [8, 25, 34] for more properties of σe(T ). Note that, if A is a self-adjoint operator on a Hilbert space, then σe(T ) = σ(T )\σd(T ). In particular, the essential spectrum is empty if and only if σ(T ) = σd(T ). We point out that this point, namely the absence of the essential spectrum, has been studied in many works, e.g., [22, 27]. RIESZ PROJECTION AND ESSENTIAL S-SPECTRUM IN QUATERNIONIC SETTING3

R S S In the quaternionic setting, if T ∈B(VH ) and q ∈ σ (T )\R (where σ (T ) denote the S−spectrum of T ), then q is not an isolated point of σS(T ). Indeed, [q] := {hqh−1 : h ∈ H∗}⊂ σS(T ), see [13]. By the compactness of [q], {q} is not an isolated part of σS(T ). However, if we set

Ω := σS(T )/ ∼ where p ∼ q if and only if p ∈ [q], and ET the set of representative, then if S [q] is an isolated part of σ (T ) then q is an isolated point of ET . The ﬁrst aim of this work is to study the isolated part of the S− spectrum of a bounded right quaternionic operators and its relation with the essential S-spectrum. To begin with, we consider the Riesz projector associated with a given quaternionic operator T which was introduced in [11]. We refer to [2, 4, 9, 13] for more details on this concept. We treat the decomposition of the essential S−spectrum of T as a function of the Riesz projector, and more generally as a function of a projector which commutes with T , see Theorem 3.1. The technique of the proof is inspired from [4]. We also discuss the Riesz decomposition theorem [32, Theorem 6] in quaternionic setting. More precisely, we prove that this decomposition is unique. Motivated by this, we study the quaternionic version of the discrete S−spectrum. Following the complex formalism given in [21, 28], we show that the essential S−spectrum of a given right operator acting on right Hilbert space does not contain discrete element of the S−spectrum. The second aim of this work is to give new results concerning the Weyl and essential S-spectra in quaternionic setting. First oﬀ, Theorem 4.5 gives a description of the boundary of the Weyl S−spectrum. The proof is based on the study of the minimal modulus of the right quaternionic operators. We also deal with the particular case of the spectral theorem of essential and Weyl S−spectra. The technique of the proof is inspired from [11]. The article is organised as follows: In Section 2, we present general def- initions about operators theory in quaternionic setting. In Section 3, we discuss the question of decomposition of the essential S−spectrum. Finally, in Section 4, we provide new results of the Weyl S−spectrum.

2. Mathematical preliminaries In this section, we review some basic notions about quaternions, right quaternionic Hilbert space, right linear operator (even unbounded and deﬁne its S−spectrum), and slice functional calculus. For details, we refer to the reader [1, 11, 13, 19, 23].

2.1. Quaternions. Let H be the Hamiltonian skew ﬁeld of quaternions. This class of numbers can be written as:

q = q0 + q1i + q2j + q3k, 4 HATEMBALOUDI,SAYDABELGACEM,ANDAREFJERIBI where ql ∈ R for l = 0, 1, 2, 3 and i,j,k are the three quaternionic imaginary units satisfying i2 = j2 = k2 = ijk = −1.

The real and the imaginary part of q is deﬁned as Re(q)= q0 and Im(q)= q1i + q2j + q3k, respectively. Then, the conjugate and the usual norm of the quaternion q are given respectively by

q = q0 − q1i − q2j − q3k and |q| = qq. The set of all imaginary unit quaternions in H is denotedp by S and deﬁned as S R 2 2 2 = q1i + q2j + q3k : q1, q2, q3 ∈ ,q1 + q2 + q3 = 1 . n o The name imaginary unit is due the fact that, for any I ∈ S, we have I2 = −II = −|I|2 = −1. For every q ∈ H\R, we associate the unique element Im(q) I := ∈ S q |Im(q)| such that

q = Re(q)+ Iq|Im(q)|. This implies that

H = CI . I[∈S where

CI := R + IR. 2.2. Right quaternionic Hilbert space and operator. In this subsection, we recall the concept of right quaternionic Hilbert space and right linear operator (see, [1, 12, 19]). R Definition 2.1. [1] Let VH be a right vector space. The map R R h., .i : VH × VH −→ H is called an inner product if it satisfies the following properties: R (i) hf,gq + hi = hf, giq + hf, hi, for all f,g,h ∈ VH and q ∈ H. R (ii) hf, gi = hg,fi, for all f, g ∈ VH . (iii) If f ∈ H, then hf,fi≥ 0 and f = 0 if hf,fi = 0. R The pair (VH , h., .i) is called a right quaternionic pre-Hilbert space. More- R over, VH is said to be right quaternionic Hilbert space, if kfk = hf,fi R p defines a norm for which VH is complete. RIESZ PROJECTION AND ESSENTIAL S-SPECTRUM IN QUATERNIONIC SETTING5

R In the sequel, we assume that VH is complete and separable. We now recall the concept of Hilbert basis in the quaternionic case. First, we review the following proposition, the proof of which is similar to its complex version, see [19, 33].

R Proposition 2.2. Let VH be a right quaternionic Hilbert space and let F = R {fk : k ∈ N} be an orthonormal subset of VH . The following properties are equivalent: R (1) For every f, g ∈ VH , the series k∈Nhf,fkihfk, gi converges absolutely and P

hf, gi = hf,fkihfk, gi. kX∈N R (2) For every f ∈ VH , we have

2 2 kfk = |hfk,fi| kX∈N

⊥ R (3) F := f ∈ VH : hf, gi = 0 for all g ∈ F = {0}. n m o R (4) hFi := flql : fl ∈ F, m ∈ N is dense in VH . n Xl=1 o R Deﬁnition 2.3. Let F be an orthonormal subset of VH . F is said to be R Hilbert basis of VH if F veriﬁes one of the equivalent conditions of Proposi- tion 2.2.

The proof of the following proposition is the same as its complex version, see [19, 33].

R Proposition 2.4. Let VH be a right quaternionic Hilbert space. Then, R (1) VH admits a Hilbert basis. R (2) Two Hilbert basis of VH have the same cardinality. R R (3) If F is a Hilbert basis of VH , then every f ∈ VH can be uniquely decomposed as follows:

f = fkhfk,fi kX∈N R where the series k∈N fkhfk,fi converges absolutely in VH . P The quaternionic multiplication is not commutative. After wards, we R recall that if VH is a right separable quaternionic Hilbert space, we can R deﬁne the left scalar multiplication on VH using an arbitrary Hilbert basis R on VH . We refer to [19] for an explanation of this construction. Let F = R fk : k ∈ N be a Hilbert basis of VH . The left scalar multiplication on n o 6 HATEMBALOUDI,SAYDABELGACEM,ANDAREFJERIBI

R VH induced by F is deﬁned as the map R R H × VH −→ VH

(q,f) 7−→ qf = fkqhfk,fi. kX∈N

The properties of the left scalar multiplication are described in the following proposition. R Proposition 2.5. [19, Proposition 3.1] Let f, g ∈ VH and p,q ∈ H, then (1) q(f + g)= qf + qg and q(fp) = (qf)p. (2) kqfk = |q|kfk. (3) q(pf) = (qp)f. (4) hqf,gi = hf,qgi. (5) rf = fr, for all r ∈ R. (6) qfk = fkq, for all k ∈ N. R It is easy to see that (p + q)f = pf + qf, for all p,q ∈ H and f ∈ VH . In R the sequel, we consider VH as a right quaternionic Hilbert space equipped with the left scalar multiplication.

R Deﬁnition 2.6. Let VH be a right quaternionic Hilbert space. A mapping R R T : D(T ) ⊂ VH −→ VH , where D(T ) denote the domain of T , is called quaternionic right linear if T (f + gq)= T (f)+ T (g)q, for all f, g ∈ D(T ) and q ∈ H. The operator T is called closed, if the graph G(T ) := {(f,Tf) : f ∈ D(T )} R R is a closed right linear subspace of VH × VH . We call an quaternionic right operator T bounded if

R kT k := sup kTfk : f ∈ VH , kfk = 1 < +∞. n o R R The set of all bounded right operators on VH is denoted by B(VH ) and R R the identity operator on V will be denoted by I R . Let T ∈ B(V ), we H VH H denote the null space of T by N(T ) and its range space by R(T ). A closed R subspace M of VH is said to be T −invariant subspace if T (M) ⊂ M. Note that the function f 7−→ Tf − fq is not right linear, we refer to [12, 13] for this point of view. The fundamental suggestion of [12] is to deﬁne the spectrum using the Cauchy kernel series. We recall these concepts from the book [13].

R R Deﬁnition 2.7. Let T : D(T ) ⊂ VH −→ VH be a right linear operator. We 2 R deﬁne the operator Qq(T ) : D(T ) −→ VH by 2 2 Q (T ) := T − 2Re(q)T + |q| I R . q VH RIESZ PROJECTION AND ESSENTIAL S-SPECTRUM IN QUATERNIONIC SETTING7

1) The S−resolvent set of T is deﬁned as follows:

S −1 R ρ (T ) := q ∈ H : N(Qq(T )) = {0}, R(T )= H and Qq(T ) ∈B(VH ) . n o 2) The S−spectrum of T is deﬁned as: σS(T )= H\ρS(T ). 3) The point S−spectrum of T is given by S H σp (T ) := q ∈ : N(Qq(T )) 6= {0} . n o Note that the S−spectrum σS(T ) is a non-empty compact subset of H, R see [13]. Let v ∈ VH \{0}, then v is a right eigenvalue of T if T (v) is a right quaternionic multiple of v. That, is T (v)= vq where q ∈ H, known as the right eigenvalue. The set of right eigenvalues coincides with the point S−spectrum, see [19, Proposition 4.5]. Now, we recall the deﬁnition of essential S−spectrum, we refer to [29, 30] R for more details. Let VH be a separable right quaternionic Hilbert space R equipped with a left scalar multiplication. Using [19], B(VH ) is a quaternionic two-sided Banach C∗−algebra with unity, and the set of all compact R R operators K(VH ) is a closed two-sided ideal of B(VH ). We consider the natural quotient map: R R R R π : B(VH ) −→ C(VH ) := B(VH )/K(VH ) R T 7−→ [T ]= T + K(VH )

R Note that π is a unital homomorphism, see [29]. The norm on C(VH ) is given by k[T ]k = inf kA + Kk. R K∈K(VH )

R Definition 2.8. [29] The essential S−spectrum of T ∈B(VH ) is the S−spectrum R of π(A) in the Calkin algebra C(VH ). That is, S S σe (T ) := σ (π(A)). 2.3. The quaternionic functional calculus. The quaternionic functional calculus is defined on the class of slice regular function f : U −→ H for some set U ⊂ H. We recall this concept and refer to [9, 11, 13] and the references therein on the matter. Definition 2.9. A set U ⊂ H is called (i) axially symmetric if [x] ⊂ U for any x ∈ U and (ii) a slice domain if U is open, U ∩ R 6= ∅ and U ∩ CI is a domain in CI , for any I ∈ S. 8 HATEMBALOUDI,SAYDABELGACEM,ANDAREFJERIBI

Deﬁnition 2.10. Let U ⊂ H be an open set. A real diﬀerentiable function f : U −→ H is said to be left s−regular (resp. right s−regular) if for every I ∈ S, the function f satisfy 1 ∂ ∂ 1 ∂ [ f(x + Iy)+ I f(x + Iy)] = 0 resp. [ f(x + Iy) 2 ∂x ∂y 2 ∂x ∂ + f(x + Iy)I] = 0 . ∂y We denote the class of left s−regular (resp. right s−regular) by RL(U) (resp. RR(U)). We recall that RL(U) is a right H−module and RR(U) is R a left H−module. Let VH be a separable right quaternionic Hilbert space equipped with a Hilbert basic N and with a left scalar multiplication. We R recall that B(VH ) is a two-sided ideal quaternionic Banach algebra with respect to the left multiplication given by (q.T )f = gqhg,Tfi and (Tq)f = T (g)qhg,fi. gX∈N gX∈N

R S Definition 2.11. Let T ∈ B(VH ) and q ∈ ρ (T ). The left S−resolvent operator is given by −1 −1 S (q, T ) := −Q (T ) (T − qI R ), L q VH and the right S−resolvent operator is defined by −1 −1 S (q, T ) := −(T − qI R )Q (T ) . R VH q R Definition 2.12. [4, Definition 2.4] Let T ∈ B(VH ) and let U ⊂ H be an axially symmetric s−domain that contains the S−spectrum σS(T ) and such that ∂(U ∩ CI ) is union of a finite number of continuously differentiable Jordan curves for every I ∈ S. We say that U is a T −admissible open set. R Definition 2.13. Let T ∈ B(VH ), W ⊂ H be open set. A function f ∈ RL(W )(resp. RR(W ))is said to be locally left regular (resp. right regular) function on σS(T ), if there is T −admissible domain U ⊂ H such that U ⊂ W .

L R We denote by RσS (T ) (resp. RσS (T )) the set of locally left ( resp. right) regular function on σS(T ). Now, we recall the two versions of the quaternionic functional calculus. R Deﬁnition 2.14. [11, Deﬁnition 4.10.4] Let T ∈ B(VH ) and U ⊂ H be a T −admissible domain. Then,

1 −1 L (1) f(T )= SL (q, T )dqI f(q) ∀f ∈ RσS (T ) 2π Z∂(U∩CI ) and 1 −1 R (2) f(T )= f(q)dqISR (q, T ) ∀f ∈ RσS (T ) 2π Z∂(U∩CI ) RIESZ PROJECTION AND ESSENTIAL S-SPECTRUM IN QUATERNIONIC SETTING9 where dqI = −dqI. The two integrals that appear in Eqs (1) and (2) are independent of the choice of imaginary unit I ∈ S and T −admissible domain, see [11, Theorem 4.10.3]. A set σ is called an isolated part of σS(T ) if both σ and σS(T ) \ σ are closed subsets of σS(T ). R Deﬁnition 2.15. [5, 9] Let T ∈ B(VH ). Denote Uσ an axially symmetric s−domain that contains that axially symmetric isolated part σ ⊂ σS(T ) S but not any other point of σ (T ). Suppose that the Jordan curves ∂(Uσ ∩ S CI) belong to the S−resolvent set ρ (T ), for any I ∈ S. We deﬁne the quaternionic Riesz projection by

1 −1 Pσ = SL (q, T )dqI . 2π Z∂(Uσ∩CI )

Remark 2.16. [5, 9] The concept of Pσ can be given by using the right −1 S−resolvent operator SR (q, T ), that is

1 −1 Pσ = dqI SR (q, T ). 2π Z∂(Uσ∩CI )

Note that Pσ is a projection that commutes with T , see [5, Theorem 2.8]. R In the sequel, we assume that VH is a separable right quaternionic Hilbert space with inﬁnite dimensional.

3. Riesz projection and essential spectrum We recall that in [29, 30], the study of the essential S−spectrum is established using the theory of Fredholm operators. The aim of this section is to show that the essential S−spectrum does not contain discrete element R of the S−spectrum. In this regard, let VH be a separable right quaternionic R S S Hilbert space. We note that if K ∈ K(VH ), then σe (A + K)= σe (A) for all R A ∈ B(VH ). We start by showing that in general the S-spectrum does not ′ R ′ satisfy this property. We deﬁne VH = B(VH , H) and call VH the right dual R space of VH . R S S Theorem 3.1. Let T ∈ B(VH ). Then, σ (T + A) ⊂ σ (A) for all A ∈ R B(VH ) if and only if T ≡ 0. Proof. Assume that T is non-zero operator and σS(T + A) ⊂ σS(A) for all R R A ∈B(V ). Let 0 R 6= x ∈ V such that H VH H

Tx = y 6= 0 R . VH ′ First step: There exists f ∈ VH such that f(x)=1 and f(y) 6= 0. 10 HATEM BALOUDI, SAYDABELGACEM, ANDAREFJERIBI

Indeed, if x = yq for some q ∈ H, the result follows from Hahn-Banach theorem. Now assume that x and y are linearly independent. Then, there R exists a right basis Z of VH such that {x,y}⊂ Z. We consider the following map

f : u = zqz 7−→ qx + qy. zX∈Z It is clear that f is right linear and f(x)=1 and f(y) 6= 0. Second step: Take A := (x − y) ⊗ f R the rank one operator on VH given by u 7−→ (x − y)f(u). We have: (T + A)2x − 2(T + A)x = (T + A)x − 2x = −x. S R S So, 1 ∈ σ (A + T ). Since dim VH > 1, then 0 ∈ σ (A). In the sequel, we assume that x 6= y. Let q ∈ σS(A). By [13, Lemma 4.2.3], we have (f(x − y))2 − 2Re(q)f(x − y)+ |q|2 = 0 if and only if q ∈ hf(x − y)h−1 : h ∈ H\{0} . n o This implies that 1 6∈ σS(A). Indeed, if 1 is an S−eigenvelue of A, there exists h ∈ H∗ such that hf(x − y)h−1 = 1 and so f(y) = 0, contradiction. R Deﬁnition 3.2. A bounded operator S ∈B(VH ) is called a quasi-inverse of R R the operator T ∈B(VH ) if there exists K1, K2 ∈ K(VH ) such that

ST = I R − K and TS = I R − K . VH 1 VH 2 R H S Lemma 3.3. Let T ∈ B(VH ), q ∈ \σe (T ). Let Rq(T ) be a quasi-inverse of Qq(T ) and A be an operator that commute with T . Then, there exist R K ∈ K(VH ) such that

ARq(T )= Rq(T )A + K. R Proof. Since AT = T A, then Qq(T )A = AQq(T ). Let K1, K2 ∈ K(VH ) such that

R (T )Q (T )= I R − K and Q (T )R (T )= I R − K . q q VH 1 q q VH 2 Then,

ARq(T )= Rq(T )A + K RIESZ PROJECTION AND ESSENTIAL S-SPECTRUM IN QUATERNIONIC SETTING11

R where K = K1ARq(T ) − Rq(T )AK2 ∈ K(VH ). R R Theorem 3.4. Let VH be a right quaternionic space and T ∈ B(VH ). Let R P be a projector in B(V ) commuting with T and let P = I R − P . Take 1 H 2 VH 1 Tj := T Pj = PjT, j = 1, 2. Then, S S S σe (T )= σe (T1|R(P1)) ∪ σe (T2|R(P2)), where Ti|M denotes the restriction of Ti to M. S R Proof. Let q 6∈ σe (T ). Then, there exist Aq(T ) ∈ B(VH ) and K1, K2 ∈ R K(VH ) such that

A (T )Q (T )= I R − K and Q (T )A (T )= I R − K . q q VH 1 q q VH 2 R Using Lemma 3.3, we infer that there exists K3 ∈ K(VH ) such that

Aq(T )P1 = P1Aq(T )+ K3 and Aq(T )P2 = P2Aq(T ) − K3. Therefore,

(3) Aq(T )= P1Aq(T )P1 + P2Aq(T )P2 + K4, R where K = (I R − 2P )K ∈ K(V ). Now, we consider the following 4 VH 1 3 H identity 2 2 2 2 (4) Qq(T ) = (T1 − 2Re(q)T1 + |q| P1) + (T2 − 2Re(q)T2 + |q| P2).

We multiply the identity (4) by Aq(T ) on the left and on the right, we obtain 2 2 I R − K = (P A (T )P + P A (T )P + K )((T − 2Re(q)T + |q| P )) VH 1 1 q 1 2 q 2 4 1 1 1 2 2 + (P1Aq(T )P1 + P2Aq(T )P2 + K4)(T2 − 2Re(q)T2 + |q| P2)] and 2 2 I R − K = ((T − 2Re(q)T + |q| P ))(P A (T )P + P A (T )P + K ) VH 2 1 1 1 1 q 1 2 q 2 4 2 2 + ((T2 − 2Re(q)T2 + |q| P2)(P1Aq(T )P1 + P2Aq(T )P2 + K4). This leads us to conclude that

2 2 I R − K = P A (T )P (T − 2Re(q)T + |q| P ) VH 1 1 q 1 1 1 1 2 2 + P2Aq(T )P2(T2 − 2Re(q)T2 + |q| P2)+ K5 and 2 2 I R − K = (T − 2Re(q)T + |q| P )P A (T )P VH 2 1 1 1 1 q 1 2 2 + (T2 − 2Re(q)T2 + |q| P2)P2Aq(T )P2 + K6, where R R K5 = K4Qq(T ) ∈ K(VH ) and K6 = Qq(T )K4 ∈ K(VH ) Take

Aq,j(T )= PjAq(T )Pj, for j = 1, 2. 12 HATEM BALOUDI, SAYDABELGACEM, ANDAREFJERIBI

Then, 2 2 P (I R − K − K )P = A (T )(T − 2Re(q)T + |q| P ) 1 VH 1 5 1 q,1 1 1 1 and 2 2 P (I R − K − K )P = (T − 2Re(q)T + |q| P )A (T ). 1 VH 2 6 1 1 1 1 q,1 2 As a consequence, Aq,1(T )|R(P1) is a quasi-inverse of (T1 − 2Re(q)T1 + 2 2 |q| P1)|R(P1). Similarly, we have Aq,2(T )|N(P1) is a quasi-inverse (T2 − 2 2Re(q)T2 + |q| P2)|N(P1). Therefore, we conclude that H S S q ∈ \(σe (T1 |R(P1)) ∪ σe (T2 |R(P2))).

S S Conversely, let q 6∈ σe (T1|R(P1))∪σe (T2|R(P2)). Then, there exists Aq,i(T ) ∈ B(R(Pi)) and Ki,j ∈ K(R(Pi)), for i, j = 1, 2, such that I I Aq,iQq(Ti |R(Pi))= R(Pi) − K1,i and Qq(Ti)Aq,i(T )= R(Pi) − K2,i. Let us deﬁne the operator

Bq(T ) := P1Aq,1(T )P1 + P2Aq,2(T )P2 We get:

Bq(T )Qq(T )= P1Aq,1(T )Qq(T )P1 + P2Aq,2(T )Qq(T )P2 I I = P1( R(P1) − K1,1)P1 + P2( R(P2) − K1,2)P2 = P1 + P2 − P1K1,1P1 − P2K1,2P2

= I R − K VH 6 R where K6 = P1K1,1P1 + P2K1,2P2 ∈ K(VH ). By a similar argument, we obtain

Q (T )B (T )= I R − K , q q VH 7 R where K7 = P1K2,1P1 + P2K2,2P2 ∈ K(VH ). Therefore, Bq(T ) is a quasi- S inverse of Qq(T ). We deduce that q 6∈ σe (T ). The result of [32, Theorem 6] can be reformulate as follows: R S Theorem 3.5. Let T ∈B(VH ) and let σ be an isolated part of σ (T ). Put R R R R R R V1,H = R(Pσ) and V2,H = N(Pσ). Then, VH = V1,H ⊕ V2,H, the spaces V1,H R and V2,H are T −invariant subspaces and S S S (5) σ (T | R )= σ and σ (T | R )= σ (A)\σ. V1;H V2,H

In the next result, we discuss the uniqueness of the decomposition (5). R R Theorem 3.6. Let T ∈ B(VH ) and P be a projection in B(VH ) such that T P = P T . Set

T := T P | and T := T (I R − P )| . 1 R(P ) 2 VH N(P ) RIESZ PROJECTION AND ESSENTIAL S-SPECTRUM IN QUATERNIONIC SETTING13

S S If dist(σ (T1),σ (T2)) > 0, then

S S R(P )= R(Pσ (T1)) and N(P )= N(Pσ (T1)) Proof. First and foremost, we have by [4, Theorem 4.4] S S S σ (T )= σ (T1) ∪ σ (T2) S S Take σ1 = σ (T1) and σ2 = σ (T2) and assume that dist(σ1,σ2) > 0. According to the proof of [32, Theorem 6], there exist a pair of a non-empty disjoint axially symmetric domains (Uσ1 ,Uσ2 ) such that

σj ⊂ Uσj , U σ1 ∩ U σ2 = ∅ C and the boundary ∂(Uσj ∩ I ) is the union of ﬁnite number of continuously diﬀerentiable Jordan curves for j = 1, 2 and for all I ∈ S. In this way, we see that Uσj is Tj-admissible open set for j = 1, 2. So by the quaternionic functional calculus, we deduce that 1 −1 I SL (s, Tj)dsI = V R for j = 1, 2 2π Z C j,H ∂(Uσj ∩ I ) and

1 −1 SL (s, Tj)dsI = 0 for i 6= j 2π Z C ∂(Uσi ∩ I ) where R R V1,H := R(P ) and V2,H := N(P ). Deﬁne the operators:

IR(P ) 0 0 0 R1 = and R2 = . 0 0 0 IN(P ) We have

1 −1 Pσ1 = SL (s, T )dsI 2π Z C ∂(Uσ1 ∩ I ) 1 −1 = dsI SR (s, T ) 2π Z C ∂(Uσ1 ∩ I ) 1 −1 1 −1 I = dsI SR (s, T )P + dsI SR (s, T )( V R − P ) 2π Z C 2π Z C H ∂(Uσ1 ∩ I ) ∂(Uσ1 ∩ I ) 1 −1 1 −1 I = dsI SR (s, T1)P + dsISR (s, T2)( V R − P ) 2π Z C 2π Z C H ∂(Uσ1 ∩ I ) ∂(Uσ1 ∩ I ) 1 −1 = dsI SR (s, T1)P 2π Z C ∂(Uσ1 ∩ I )

= Pσ1 P

= (I − Pσ2 )P = R1. 14 HATEM BALOUDI, SAYDABELGACEM, ANDAREFJERIBI

We conclude that

R(P )= R(Pσ1 ). By the similar arguments, we achieve that

Pσ2 = R2 and so, N(P )= R(Pσ2 )= N(Pσ1 ). We now analyze the Riesz projection associated with the isolated spheres. To start off, we give the following definition: R S Definition 3.7. T ∈ B(VH ). A point q ∈ σ (T ) is called an eigenvalue R R R of finite type if VH is a direct sum of T −invariant subspaces V1,H and V2,H such that R (H1) dim(V1,H) < ∞, S S (H2) σ (T | R ) ∩ σ (T | R )= ∅, V1,H V2,H S (H3) σ (T | R ) = [q]. V1,H Remark 3.8.

(1) In complex spectral theory, in a complex Hilbert space VC, for a continuous linear operator, T , the condition (H3) is replaced by

σ(T |V1,C )= {q} (where VC is a direct sum of T −invariant subspaces V1,C and V2,C), see [21]. In the quaternionic setting, we must take the S S whole 2−sphere [q] because if q ∈ σ (T | R ), then [q] ⊂ σ (T | R ). V1,H V1,H R S (2) Let T ∈ B(VH ). If q ∈ σ (T )\R, then q is not an isolated point of σS(T ). Take Ω := σS(T )/ ∼

where p ∼ q if and only if p ∈ [q]. Let ET denote the set of representatives of Ω. R (3) By [24, Proposition 4.44 and Theorem 4.47], If dim(VH ) < ∞ , then the S−spectrum of T consists of right eigenvalues only and ♯ET < ∞. In particular, if T satisﬁed the assumptions (H1) and S (H3), then q ∈ σp (T ). R Lemma 3.9. Let T ∈ B(VH ). Let ET denote the set of representatives as above. Then, q is an isolated point of ET if and only if [q] is an isolated part of σS(T ). S Proof. Assume that [q] is isolated part of σ (T ) and let Uq ⊂ H be an open set such that S [q]= σ (T ) ∩ Uq.

If there exists q 6= p ∈ ET ∩ Uq, then p 6∈ [q]. This implies that

ET ∩ Uq = {q}. RIESZ PROJECTION AND ESSENTIAL S-SPECTRUM IN QUATERNIONIC SETTING15

Conversely, if q is an isolated point of ET , then [q] is an open subset of σS(T ). Since [q] is compact, then [q] is isolated part of σS(T ).

Let VC be a complex Hilbert space and T be a continuous linear operator acting on VC. It follows from [21, Theorem 1.1] that if q ∈ σ(T ) (where σ(T ) denotes the complex spectrum of T ), then q is an eigenvalue of finite type if and only if dim R(P{q}) < ∞, where P{q} is the Riesz projection corresponding to the isolated point q. In the next theorem, we show that the same is true for the right eigenvalue of finite type in the quaternionic setting. R S Theorem 3.10. Let T ∈ B(VH ) and q ∈ σp (T ). Then, q is a right eigenvalue of finite type if and only if {q} is an isolated part in ET and dim R(P[q]) < ∞. Proof. Assume that q is a right eigenvalue of finite type and consider the direct sum R R R VH = V1,H ⊕ V2,H with the properties (H1) − (H3). By Proposition 3.6, this decomposition is unique and so R V1,H = R(P[q]). The converse comes from Theorem 3.5. R Corollary 3.11. Let T ∈B(VH ). Then, S S S σe (T ) ⊂ σ (T )\σd (T ) S where σd (T ) denotes the set of all right eigenvalue of finite type. Proof. Assume that q is a right eigenvalue of finite type. Then, [q] is an S isolated part of σ (T ) and dim R(P[q]) < ∞. Therefore, S dim R(T P[q]) < ∞ and so σe (T |R(P[q]))= ∅. Using Theorem 3.4, we infer that S S I σe (T )= σe (T |R( R −P[q])). VH On the other hand, S S S I I σe (T |R( R −P[q])) ⊂ σ (T |R( R −P[q]))= σ (T )\[q]. VH VH S This leads to conclude that q 6∈ σe (T ). Example 3.12. We consider the right quaternionic Hilbert space: 2 2 2 ℓH(Z) := x : Z −→ H such that kxk := |xi| < ∞ . n Xi∈Z o with the right scalar multiplication

xa = (xia)i∈Z 16 HATEM BALOUDI, SAYDABELGACEM, ANDAREFJERIBI for x = (xi)i∈Z and a ∈ H. The associated scalar product is given by

hx,yi := hx,yi 2 := x y . ℓH(Z) i i Xi∈Z The right shift is the map: 2 2 T : ℓH(Z) −→ ℓH(Z)

x 7−→ y = (yi)i∈Z where yi = xi+1 if i 6= −1 and 0 if i = −1. We have 2 2 2 kT (x)k = |xi| ≤ kxk . iX6=−1 The S−spectrum of T was studied in [7, 31]. In particular, we have S S σ (T )= σp (T )= ∇H(0, 1), where ∇H(0, 1) is the closed quaternionic unit ball. By Theorem 3.10, none of these S−eigenvalues are of ﬁnite type. We recall: R Lemma 3.13. [18, Corollary 2.22] Let VH be a quaternionic right vector R space and E be a right linear independent subspace of VH . Then, there exists R a right basis B of VH such that E ⊂ B. In particular, every quaternionic right vector space has a right basis. R Proposition 3.14. [24, Proposition 4.44] Let VH be a quaternionic Hilbert R space and T ∈B(VH ). If q1, ...., qn ∈ H are right eigenvalues of T such that R [qi] 6= [qj], ∀1 ≤ i < j ≤ n (n ≥ 2), and Qqj (T )xj = 0, 0 6= xj ∈ VH , ∀1 ≤ R j ≤ n, then x1,x2, ..., xn are right-linearly independent in VH .

We have the following lemma: R R Lemma 3.15. Assume that dim(VH ) < ∞ and let T ∈ B(VH ). Then, ♯ET < ∞. In this case, set

ET = q1,q2, ..., qn , n o R R R R then VH is a direct sum of T −invariant right subspaces V1,H,V2,H, ...., Vn,H. R R Moreover, if Ti := T | R : V H −→ V H, for i = 1, ..., n, then Vi,H i, i,

S −1 ∗ σ (Ti)= hqih : h ∈ H . n o Proof. Combine Lemma 3.13, Proposition 3.14 and Theorem 3.10 R R Let VH be a quaternionic right Hilbert space, T ∈B(VH ) and q be a right eigenvalue of T of ﬁnite type. In this way, we see that R VH = R(P[q]) ⊕ R(PσS (T )\[q]) RIESZ PROJECTION AND ESSENTIAL S-SPECTRUM IN QUATERNIONIC SETTING17

Deﬁnition 3.16. The algebraic multiplicity of the right eigenvalue q is, by deﬁnition, the dimension of the space R(P[q]). In the next, we write:

mT (q) := dim R(P[q]). We refer to [21, 25] for the definition in complex setting. Now, inspired by the description of the Riesz projection in the complex setting, see [21], we give a quaternionic version of a result describing the finite part of right eigenvalues of finite type.

R R Theorem 3.17. Let VH be a quaternionic Hilbert space, T ∈ B(VH ) and S σ be an isolated part of σ (T ). Set Pσ the Riesz projection correspond to σ and take σ ∼ ET = σ/ = where q =∼ p if and only if p ∈ [q]. Then, dim R(Pσ) < ∞ if and only if σ σ ♯ET < ∞ and q is a right eigenvalue of ﬁnite type for all q ∈ ET . Besides, if so, then

dim R(Pσ)= dim R(P[q]). σ qX∈ET

Proof. If dim R(Pσ) < ∞, then PσT is a ﬁnite rank operator. In this way, we see that ♯Eσ = n< ∞. T |R(Pσ) By Lemma 3.15, R R R R(Pσ)= V1,H ⊕ V2,H ⊕ ... ⊕ Vn,H R where Vj,H is T −invariant with the properties S σ (T | R ) = [qj], j=1,...,n. Vj,H

Let i ∈ {1, ..., n}. Since Pσ is a projection, then R VH = N(Pσ) ⊕ R(Pσ). In this fashion, we have R R R VH = Vi,H ⊕ Wi,H where R R R R R Wi,H = V1,H ⊕ ... ⊕ Vi−1,H ⊕ Vi+1,H ⊕ ... ⊕ Vn,H ⊕ N(Pσ). R R S As consequence, V H and W H are T −invariant and σ (T | R ) = [qi]. So, i, i, Vi,H qi is a right eigenvalue of ﬁnite type. 18 HATEM BALOUDI, SAYDABELGACEM, ANDAREFJERIBI

σ Conversely, set ET = {q1, q2..., qn}, where qi is a right eigenvalue of T of ﬁnite type for all i ∈ {1, 2, ..., n}. Applying [12, Theorem 5.6], we have n I R(Pσ ) = P[qi]|R(Pσ ). Xi=1

Since P[qi]P[qj ] = 0 for all i 6= j, then

R(Pσ)= R(P[q1]) ⊕ R(P[q2]) ⊕ ... ⊕ R(P[qn]). This implies that n

dim(R(Pσ)) = dim(R(P[qi])). Xi=1

In particular, we have Pσ which is a ﬁnite rank operator. Remark 3.18. In the complex spectral theory, much attention has ben paid to eigenvalue of ﬁnite type, see [10, 21, 25, 26, 28]. It is useful for the study of the essential spectrum of certain operators-matrices. We refer to [10] for this point on the two-groupe transport operators. More precisely, let VC be a complex Banach space and let T be a closed operator in VC. The Browder resolvent set of T is given by

ρB(T ) := ρ(T ) ∪ σd(T ), where we use the notation ρ(.) for the resolvent set of T and σd(.) the set of eigenvalues of ﬁnite type of T . In fact, the usual resolvent −1 Rλ(A) := (A − λ) can be extended to ρB(T ), e.g. [28]. Motivated by this, [10] gives a version of the Frobenius-Schur factorization using the Browder resolvent. This makes it possible to study the essential spectrum of serval types operators-matrices. In this paper, we have described the discrete S−spectrum in quaternionic setting. In this regard, as in complex case, we can deﬁne the spherical Browder resolvent. Although, we avoided studying it in this paper, we will cover that in a future article.

4. Some results on the Weyl S-spectrum In this section, we develop a deeper understanding of the concept of the Weyl S-spectrum of the bounded right linear operator. More precisely, we describe the boundary of the S-spectrum. Likewise, we deal with the particular case of the spectral theorem. To begin with, we recall: R Deﬁnition 4.1. [30] Let T ∈B(VH ). The Weyl S−spectrum is the set

S S σW (T )= σ (T + K). R K∈K\(VH ) RIESZ PROJECTION AND ESSENTIAL S-SPECTRUM IN QUATERNIONIC SETTING19

The study of the essential and the Weyl S−spectra are established using the Fredholm theory, see [29, 30]. We refer to [7] for the investigation of the Fredholm and Weyl elements with respect to a quaternionic Banach algebra homomorphism. R Deﬁnition 4.2. A Fredholm operator is an operator T ∈ B(VH ) such that R R N(T ) and VH /R(T ) are ﬁnite dimensional. We will denote by Φ(VH ) the set of all Fredholm operators.

From [29, 30], we have R R R Φ(VH )=Φl(VH ) ∩ Φr(VH ) where R R Φl(VH )= T ∈B(VH ) : R(T) is closed and dim(N(T )) < ∞ n o and R R † Φr(VH )= T ∈B(VH ) : R(T) is closed and dim(N(T )) < ∞ . n o R R Let T ∈ Φl(VH ) ∪ Φr(VH ). Then, the index of T is given by R i(T ) := dim N(T ) − dim(VH /R(T )).

R Theorem 4.3. [29, 30] Let T ∈B(VH ). Then, S H S H σe (T )= \ΦT and σW (T )= \WT where R ΦT := q ∈ H : Qq(T ) ∈ Φ(VH ) n o and R WT := q ∈ H : Qq(T ) ∈ Φ(VH ) and i(Qq(T )) = 0 . n o R R Remark 4.4. Let VH be a quaternionic space and T ∈B(VH ). (1) Note that, in general, we have S S S S S σe (T ) ⊂ σW (T )= σ1,W (T ) ∪ σ2,W (T ) ⊂ σ (T )\σd(T ). where S H H R σ1,W (T ) := \ q ∈ : Qq(T ) ∈ Φl(VH ) and i(Qq(T )) ≤ 0 n o and S H H R σ2,W (T ) := \ q ∈ : Qq(T ) ∈ Φr(VH ) and i(Qq(T )) ≥ 0 . n o S In particular, σW (T ) does not contain eigenvalues of ﬁnite type. (2) In [7], one proves that q 7−→ i(T ) is constant on any component of ΦT . In this way, we see that if ΦT is connected, then S S σe (T )= σW (T ). 20 HATEM BALOUDI, SAYDABELGACEM, ANDAREFJERIBI

The ﬁrst result in this section is the next theorem. R Theorem 4.5. Let T ∈B(VH ). Then, S S ∂σW (T ) ⊂ σ1,W (T ).

In particular, if ΦT is connected, then S H R ∂σe (T )= ∂σW (T )= q ∈ : Qq(T ) 6∈ Φl(VH ) . n o To prove Theorem 4.5, we ﬁrst study the concept of the minimum mod- R R ulus. Let VH be a separable right Hilbert space and T ∈ B(VH ). The minimum modulus of T is given by µ(T ) := inf kTxk. kxk=1

To begin with, we give the following lemma. Lemma 4.6. Let T and S be two bounded right linear operators on a right quaternionic Hilbert space. Then, (1) If kT − Sk < µ(T ), then µ(S) > 0 and R(S) is not a proper subset of R(T ).

µ(T ) (2) If kT − Sk < 2 , then R(S) is not a proper subset of R(T ) and R(T ) is not a proper subset of R(S). Proof. The proof is the same as for the complex Banach space, see Lemma 2.3 and lemma 2.4 in [20] for a complex proof.

R For T ∈B(VH ), q ∈ H and ε> 0 we set:

O(T,q,ε) := q′ ∈ H : 2|Re(q) − Re(q′)|kT k + ||q′|2 − |q|2| < ε. n o It is clear that O(T,q,ε) is an open set in H. R S S Corollary 4.7. Let T ∈B(VH ) and q0 ∈ ρ (T ). Then, q ∈ ρ (T ) for each q ∈ O(T,q,µ(Qq0 (T ))).

Proof. Let q ∈ O(T,q,µ(Qq0 (T ))). Then,

2 2 kQq(T ) − Qq0 (T )k = k2(Re(q0) − Re(q))T + |q| − |q0| k 2 2 ≤ 2|(Re(q0) − Re(q)|kT k + ||q| − |q0| |

<µ(Qq0 (T )). We can apply Lemma 4.6 to conclude that

R µ(Qq(T )) > 0 and R(Qq(T )) = R(Qq0 (T )) = VH . S By [29, Proposition 3.5], R(Qq(T )) is closed. Hence, q ∈ ρ (T ). We recall: RIESZ PROJECTION AND ESSENTIAL S-SPECTRUM IN QUATERNIONIC SETTING21

Lemma 4.8. [13, Lemma 7.3.9] Let n ∈ N and q,s ∈ H. Set 2n n n n 2 P2n(q)= q − 2Re(s )q + |s | . Then, 2 2 P2n(q)= Q2n−2(q)(q − 2Re(s)q + |s| ) 2 2 = (q − 2Re(s)q + |s| )Q2n−2(q), where Q2n−2(q) is a polynomial of degree 2n − 2 in q.

Proof of Theorem 4.5 Set: f(T ):= sup µ(T + K). R K∈K(VH ) Similar proof in the complex case, we have f(T ) > 0 if and only if R R T ∈ Φl(VH ) and dim N(T ) ≤ dim(VH /R(T )). S S Now, Since σe (T ) is not empty (e.g., [29, Proposition 7.14]) and σe (T ) ⊂ S S S σW (T ), then ∂σW (T ) is not empty. Let us then take an element p in ∂σW (T ). S Assume that p 6∈ σ1,w(T ). Then,

f(Qp(T )) > 0. R So, there exists K0 ∈ K(VH ) such that

µ(Qp(T )+ K0) > 0.

µ(Qp(T )+K0) Since O(T,p, 2 ) is an open neighborhood of p and

R p ∈ q ∈ H : Qq(T ) ∈ Φ(VH ) and i(Qq(T )) = 0 , n o µ(Qp(T )+K0) then there exists p0 ∈ O(T,p, 2 ) such that

p0 ∈ WT . On the other hand, 2 2 kQp0 (T )+ K0) − Qp(T ) − K0)k ≤ ||p0| − |p| | + 2|Re(p) − Re(p0)|kT k µ(Q (T )+ K ) < p 0 . 2 Applying Lemma 4.6, we obtain R R(Qp(T )+ K0)= VH .

Indeed, since µ(Qp0 + K0) > 0 and p0 ∈ WT , then R dim(VH /R(Qp0 (T )+ K0)) = 0. In this way, we see that R Qp(T )+ K0 ∈ Φ(VH ) and i(Qp(T )+ K0) = 0. S This implies that, p 6∈ σW (T ). 22 HATEM BALOUDI, SAYDABELGACEM, ANDAREFJERIBI

The rest of the proof follows immediately from [7, Theorem 5.13]. We will now deal with the particular spectral theorem for the essential and Weyl S-spectra. R Theorem 4.9. Let T ∈B(VH ). Then, S n n H S S n S n S n σe (T )= q ∈ : q ∈ σe (T ) = (σe (T )) and σW (T ) ⊂ (σW (T )) . n o Proof. According to [13, Lemma 3.10] and the proof of [13, Theorem 7.3.11] we have n−1 2n n 2 2 2 T − 2Re(q)T + |q| I R = (T − 2Re(q )T + |q | I R ), VH j j VH jY=0 n where qj, j = 0, ..., n − 1 are the solutions of p = q in the complex plane C S n n R Iq . Let q 6∈ σe (T ). Then, Qq(T ) ∈ Φ(VH ). Since

Qqj (T )Qqi (T )= Qqi (T )Qqj (T ) for all i, j = 0, ..., n − 1. We can apply [29, Theorem 6.13], we infer that R Qqi (T ) ∈ Φ(VH ) for all i = 0, ..., n − 1. n S n In this way, we see that q = λi 6∈ (σe (T )) . To prove the inverse inclusion, S n we consider q 6∈ (σe (T )) . Then, R Qqi (T ) 6∈ Φ(VH ) for all i = 0, ..., n − 1. R S n So, by [29, Corollary 6.14], Qq(T ) ∈ Φ(VH ) and hence q 6∈ σe (T ). Now, S n S n S n we still need to prove that σW (T ) ⊂ (σW (T )) . Let q 6∈ (σW (T )) . Then,

S qi 6∈ σW (T ), for all i = 0, ..., n − 1. Using Theorem 4.3, we have R Qqi (T ) ∈ Φ(VH ) and i(Qqi (T )) = 0 for all i = 0, ..., n − 1. Again applying [29, Theorem 6.13], n−1 n R n Qq(T ) ∈ Φ(VH ) and i(Qq(T )= i(Qqi (T )) = 0. Xi=0 S n So, q 6∈ σW (T ). References [1] Adler S.L.: Quaternionic quantum mechanics and quantum ﬁelds. New york: The clarendon Press., oxford university press; 1995. (international series of monographs on physics; vol 88). p. xii+586. [2] Alpay D., Colombo F., Gantner J., Sabadini I.: A new resolvent equation for the S-functional calculus, J. Geom. Anal. 2015;25:1939-1968. [3] Alpay D., Colombo F., Sabadini I.: Slice hyperholomorphic Schur analysis. Springer international publisching, 2016. RIESZ PROJECTION AND ESSENTIAL S-SPECTRUM IN QUATERNIONIC SETTING23

[4] Alpay D., Colombo F., Sabadini I.: Krien-Langer factorization and related topics in the slice hyperholomorphic setting, J. Geom. 2014;24(2):843-872. [5] ALpay D., Colombo F., David P.K.: The spectral theorem for unitary operators based on the S-spectrum, Milan J. Math. 2016;84:41-61. [6] Alpay D., Colombo F., Kimsey D.P.: The spectral theorem for quaternionic unbounded normal operators based on the S-spectrum, J. Math. Phys., 2016; 57(2), 023503,27 pp. [7] Baloudi H.: Fredholm theory in quaternionic Banach algebra, arXiv preprint arXiv:2012.11032 (2020). [8] Baloudi H., Jeribi A.: Left-right Fredholm and Weyl spectra of the sum of two bounded operators and applications. Mediterr. J. Math. 2014;11(3):939-953. [9] Cerejeiras P., Colombo F., Kähler U., Sabadini I.: Perturbation of normal operators, Trans. Amer. Math. Soc., 2019;372(5):3257-32-81. [10] Charfi S., Jeribi A.: On a caracterization of the essential spectra of some matrix-operators and application to two-group transport operators. Math. Z. 2009; 262(4):775-794. [11] Colombo F., Sabadini I., Struppa D.C.: Noncommutative Functional Calculus- Theory and Applications of Slice Hyperholomorphic Functions. Vol. 289. Progress in mathematics. Basel: Birkäuser, (2011). [12] Colombo F., Gentili G., Sabadini I., Struppa D.C.: Non commutative functional calculus: bounded operators. Complex Anal. Oper. Theory. 2010;4:821-843. [13] Colombo F., Gantner J., Kimseyn D.P.: Spectral theory on the S-spectrum for quaternionic operators. Operator Theory: Advance and Application, 270. Birkhäuser/Springer, Cham, 2018. ix+356 pp. [14] Colombo F., Gantner J.: Quaternionic closed operators, fractional powers and fractional diffusion processes, Operator Theory: Advances and Applications, 274. Birkhäser/ Springer, Cham, 2019. [15] Colombo F., Jonathan G., Stefano P.: An introduction to hyperholomorphic spectral theories and fractional powers of vector operators. Adv. Appl. Clifford Algebr. 2021;31(3):1-37. [16] Colombo F., Sabaini I., Sommen F.: The Fueter mapping theorem in integral form and the F-functional calculus. Math. Methods Appl. Sci. 2010;33(17):2050- 2066. [17] Colombo F., Sabadini I., Struppa D.C.: A new functional calculus for non- commuting operators. J. Funct. Anal. 2008;254(8):2255-2274. [18] Ganter J.: Slice hyperholomorphic functions and the quaternionic functional calculus. Ph. D. Thesis, Vienna University of Technology 2014. [19] Ghiloni R., Moretti V., Perotti A.: Continuous slice functional calculus in quaternionic Hilbert spaces. Rev. Math. Phys. 2013;25(4):1350006, 83 pp. [20] Gindler H.A.: Taylor AE. The minimum modulus of a linear operator and it’s use in spectral theory. Stud. Math. 1962;22:15-41. [21] Gohberg I.C., Golberg S., Kaashoek M.A.: Classes of Linear Operators, Vols. I,II, Operator Theory: Advances and Applications, Birkhäuser, 1990,1993. [22] Golénia S.: Hardy inequality and asymptotic eigenvalue distribution for discrete Laplacian, J. Func. Anal. 2014: 266(5):2662-2688. [23] Hamilton,W.R: Eelement of quaternians.Part1, reprint of the 1866 orig- inal.Cambridge Library Collection, Cambridge University Press, Cam- bridge(2009). [24] Hang K., Wang M.: Slice regular weighted composition operators, Complex Variable and Elliptic Equation 2020:1-62. [25] Jeribi A.: Spectral theory and applications of linear operators and block operator matrices. New York: Springer-Verlag; 2015. 24 HATEM BALOUDI, SAYDABELGACEM, ANDAREFJERIBI

[26] Jeribi A.: Linear operators and their essential pseudospectra. Boca Raton: CRC Press; 2018 [27] Keller M.: The essential spectrum of the Laplacian on rapidly branching tessel- lations, Math. Ann. 2010; 346(1):51-66. [28] Lutgen J.: On essential spectra of operator-matrices and their Feshbach maps, J. Math. Anal. Appl. 2004;289:419-430. [29] Muraleetharam B., Thirulogasanthar K.: Fredholm operators and essential S- spectrum in the quaternionic setting. J. Math. Phys. 2018;59(10):103506, 27p. [30] Muraleetharam B., Thirulogasanthar K.: Weyl and Browder S-spectra in a right quaternionic Hilbert space. J. Geom. Phys. 2019;135:7-20. [31] Muraleetharam B., Thirulogasanthar K.: Decomposable operators, local S- spectrum and S-spectrum in the quaternionic setting. arXiv preprint arXiv: 190505936 (2019). [32] Pamula S.K.: Strongly irreducible factorisation of quaternionic operators and Riesz decomposition theorem, Banach J. Math. Anal. 2021:15(1), 25p. [33] Viswanath K.: Normal operators on quaternionic Hilbert space. Trans. Amer. Math. Soc. 1971;162:337-350. [34] Wolf F.: On the invariance of the essential spectrum under a change of the boundary conditions of partial diﬀerential operators, Indag. Math. 1959; 21:142- 147.

Hatem Baloudi, Department of Mathematics, Faculty of Sciences of Gafsa, University of Gafsa, 2112 Zarroug, Tunisia., E-mail adress: [email protected]; [email protected]

Sayda Belgacem, Departement of Mathematics Faculty of Sciences of Sfax, University of Sfax, Route de Soukra km 3.5, B.P. 1171, 3000 Sfax, Tunisia., E- mail adress: [email protected]

Aref Jeribi, Departement of Mathematics Faculty of Sciences of Sfax, University of Sfax, Route de Soukra km 3.5, B.P. 1171, 3000 Sfax, Tunisia., [email protected]