Chapter 2 Unbounded Operators and Abstract Functional Calculus

2.1 Unbounded Operators

We start with an introduction to unbounded operators and their spectral theory. This will be brief and cursory so that we can quickly pass on to more substantial material. For a more detailed account the reader is referred to Appendix A.4. By an (unbounded) operator from a Banach space X to a Banach space Y we mean a linear mapping A : dom(A) → X where dom(A) is a linear subspace of X. An operator is closed if one has the implication

dom(A) 3 xn → x ∈ X, Axn → y ∈ Y ⇒ x ∈ dom(A), Ax = y.

Equivalently, A is closed if dom(A) endowed with the graph norm

kxkA := kxkX + kAxkY (x ∈ dom(A)) is a Banach space. The set of all closed operators is denoted by

C(X; Y ) (and C(X) := C(X; X)).

We reserve the term bounded operator for the elements of L(X; Y ). By the closed graph theorem, an operator A is bounded if and only if it is closed and satisfies dom(A) = X (i.e., it is fully defined). Operators A can be identified with their graph

graph(A) = {(x, y) ∈ X ⊕ Y : x ∈ dom(A), Ax = y}, which is a linear subspace of X ⊕ Y (a so-called linear relation). The operator is closed if and only if its graph is closed in X ⊕ Y . Set theory purists would claim that an operator actually is the same as its graph. So we are justiﬁed to identify these concepts, and will do so henceforth. We use the

25 26 2 Unbounded Operators and Abstract Functional Calculus notation “Ax = y” synonymously with (x, y) ∈ A, so that “x ∈ dom(A)” is tacit here. Identifying operators with (certain) linear relations gives immediate mean- ing to the inclusion statement “A ⊆ B”. Equivalently, this can be expressed through the implication

Ax = y ⇒ Bx = y (x ∈ X, y ∈ Y ) or through: dom(A) ⊆ dom(B) and B|dom(A) = A. An operator A is closable if there is a closed operator B such that A ⊆ B. In this case, there is a smallest closed operator containing A, named the closure of A and denoted by A. (This is consistent with the interpretation as linear subspaces of X ⊕Y .) A subspace D ⊆ dom(A) is a core for a closed operator A, if A is the closure of A|D. Algebra for (unbounded) operators is more complicated than for bounded ones due to domain issues. For example, the sum A + B of two operators is deﬁned by

dom(A + B) := dom(A) ∩ dom(B), (A + B)x := Ax + Bx.

The composition AB of two operators (when meaningful) is deﬁned by

ABx = z ⇐⇒def. ∃ y : Bx = y ∧ Ay = z.

Sum and product are (trivially) associative, but distributivity fails in general. The most important fact to remember about products is: If A is closed and B is bounded, then AB is closed. (See Exercise 2.2.) For an operator A, its inverse A−1 is the linear relation deﬁned by

(y, x) ∈ A−1 ⇐⇒def. Ax = y.

The inverse A−1 is an operator if and only if A is injective. Obviously, A−1 is closed if and only if A is closed. An operator A is invertible if A−1 is a bounded operator. Spectral Theory. The notions of spectrum and resolvent (set) for general operators A on a Banach space X are exactly the same as for bounded operators. The resolvent set is

ρ(A) := {λ ∈ C : λ − A is invertible}, where λ − A := λI − A. The resolvent is the mapping

R(·,A): ρ(A) → L(X),R(λ, A) := (λ − A)−1. 2.2 Multiplication Operators 27

The spectrum is σ(A) := C \ ρ(A), but it is sometimes helpful to think of ∞ as a spectral value of an operator which is not bounded. (Actually, for spectral theory it is much more convenient to work not just with operators but with general linear relations, see Appendix A.4.) As for bounded operators, the resolvent set is open and the resolvent is holomorphic and satisfies the resolvent identity. Differentiating yields d 2 dz R(z, A) = −R(z, A) and the usual power series representation obtains (see Corollary A.15). Eigenvalues and approximate eigenvalues and the corresponding notions of point spectrum σp(A) and approximate point spectrum σa(A) are defined as for bounded operators, see Lemma A.17 and A.18.

2.2 Multiplication Operators

Let Ω = (Ω, Σ, µ) be any measure space. For an essentially measurable function a : Ω → C we deﬁne the corresponding multiplication operator Ma on Lp(Ω), 1 ≤ p ≤ ∞, by

p p Maf := af for f ∈ dom(Ma) := {g ∈ L (Ω) : ag ∈ L (Ω)}.

The next theorem collects the relevant properties. We restrict to semi-ﬁnite measure spaces for convenience (see Appendix A.1).

Theorem 2.1. Let Ω = (Ω, Σ, µ) be a semi-ﬁnite measure space. Then for p a measurable function a : Ω → C the multiplication operator Ma on L (Ω), 1 ≤ p < ∞, has the following properties:

a) Ma is a closed operator. −1 1 b) Ma is injective if and only if µ[ a = 0 ] = 0. In this case, Ma = Ma−1 .

c) Ma is densely deﬁned. ∞ d) Ma is bounded if and only if a ∈ L (Ω). In this case kMak = kakL∞ .

e) If b : Ω → C is measurable, then MaMb ⊆ Mab and Ma + Mb ⊆ Ma+b. Moreover, dom(MaMb) = dom(Mb) ∩ dom(Mab). f) If b ∈ L∞(Ω) then

MaMb = Mab and Ma + Mb = Ma+b.

g) σ(Ma) = σa(Ma) = essran(a), the essential range of a.

h) λ ∈ ρ(Ma) =⇒ R(λ, Ma) = M 1 . λ−a

1 −1 1 Here as in most cases, a is synonymous with a . 28 2 Unbounded Operators and Abstract Functional Calculus

i) Up to equality µ-almost everywhere, a is uniquely determined by Ma.

p p Proof. a) Suppose that (fn, afn) → (f, g) in L ⊕ L . Passing to a subse- quence if necessary we may suppose that fn → f almost everywhere. But then af = g. Hence f ∈ dom(Ma) and g = Maf. p b) Ma is not injective precisely when there is 0 6= f ∈ L such that af = 0, i.e. [ f 6= 0 ] ⊆ [ a = 0 ]. Since the measure is semi-finite, such a function f can be found precisely when µ[ a = 0 ] > 0. If µ[ a = 0 ] = 0 then a−1 is an essentially measurable C-valued function and af = g µ-almost everywhere if and only if f = a−1g µ-almost everywhere. −1 Hence Ma = Ma−1 as claimed. p p c) Let f ∈ L (Ω), define An := [ |a| ≤ n ] and let fn = 1An f. Then fn ∈ L p and fn → f in L . Moreover, |afn| ≤ n |f|, hence fn ∈ dom(Ma). ∞ d) It is easy to see that a ∈ L implies that Ma is bounded and kMak ≤ kakL∞ . For the converse, suppose that Ma is bounded and fix c > 0 such that µ[ |a| ≥ c ] > 0. Then by the semi-finiteness of the measure space Ω there is a set A ⊆ [ |a| ≥ c ] of strictly positive but finite measure. Applying Ma to 1A yields Z p p p p µ(A)c ≤ |a1A| = kMa1Akp ≤ kMak µ(A). Ω

It follows that c ≤ kMak, and the claim is proved. e) and f) are straightforward. g) Passing to λ − a if necessary we only need to show that

0 ∈ σ(Ma) =⇒ 0 ∈ essran(a) =⇒ 0 ∈ σa(Ma).

If 0 ∈/ essran(a) then there is ε > 0 such that µ[ |a| ≤ ε ] = 0. Hence Ma −1 −1 1 −1 is injective, and Ma = M1/a. But ka kL∞ ≤ ε < ∞ and hence Ma is bounded. It follows that 0 ∈ ρ(Ma). 1 Suppose that 0 ∈ essran(a). Then for each n ∈ N one has µ |a| ≤ n > 0. 1 By the semi-finiteness we can find a set An ⊆ |a| ≤ n of finite and strictly −1 positive measure. Let cn = k1An kp and fn := cn1An . Then kfnkp = 1 and 1 p fn ∈ dom(Ma) and |Mafn| = |af| ≤ n |f| → 0 in L . Hence, (fn)n is an approximate eigenvector for 0, and 0 ∈ σa(Ma) as claimed. h) is clear. i) Suppose that a and b are measurable functions on Ω such that Ma = Mb p p as operators on L (Ω). For n ∈ N and f ∈ L define fn := f1[ |a|+|b|≤n ]. Then fn ∈ dom(Ma) ∩ dom(Mb) and hence

a1[ |a|+|b|≤n ]f = Mafn = Mbfn = b1[ |a|+|b|≤n ]f.

It follows that a1[ |a|+|b|≤n ] = b1[ |a|+|b|≤n ] almost everywhere for each n ∈ N and hence a = b almost everywhere. 2.3 Functional Calculus for Multiplication Operators 29

Multiplication operators can be considered on other function spaces, for instance on the space Cb(Ω), where Ω is a metric space (Exercise 2.5), or on the space C0(Ω), where Ω is a locally compact space. (Here, the multiplicators a are continuous functions.)

2.3 Functional Calculus for Multiplication Operators

From now on we ﬁx a (semi-ﬁnite) measure space Ω = (Ω, Σ, µ), a µ- essentially measurable function a : Ω → C and a number 1 ≤ p < ∞ and p consider the operator A := Ma on X := L (Ω). Moreover, let us write

K := essran(a) = σ(A) ⊆ C, which is a closed subset of C. It is easy to see (Exercise 2.6) that a ∈ K µ-almost everywhere.

Equivalently, the Borel measure ν := a∗µ is supported on K:

K = essran(a) = supp(a∗µ) = supp(ν).

Hence, if f : K → C is ν-almost everywhere measurable, then f ◦a is µ-almost everywhere measurable, and we can form the closed(!) operator

Φ(f) := Mf◦a on X = Lp(Ω). The set L0(K, ν) of ν-almost everywhere measurable functions (modulo ν-a.e. null functions) on K is an algebra. The emerging mapping

Φ :L0(K, ν) → C(X) is called the functional calculus for the operator A = Ma. Here is a listing of some essential properties. Theorem 2.2. Let Ω = (Ω, Σ, µ) be a semi-ﬁnite measure space, let a ∈ 0 L (Ω) and 1 ≤ p < ∞, K := essran(a) and ν := a∗µ as above. Let, furthermore,

0 p Φ :L (K, ν) → {operators on L (Ω)},Φ(f) = Mf◦a be the associated functional calculus. Then for all f, g ∈ L0(K, ν) and λ ∈ C the following statements hold: a) The operator Φ(f) is closed. b) Φ(1) = I. c) λΦ(f) ⊆ Φ(λf) and Φ(f) + Φ(g) ⊆ Φ(f + g). 30 2 Unbounded Operators and Abstract Functional Calculus

d) Φ(f)Φ(g) ⊆ Φ(fg) with

dom(Φ(f)Φ(g)) = dom(Φ(g)) ∩ dom(Φ(fg)).

e) Φ(f) is injective if and only if ν[ f = 0 ] = 0, and in this case Φ(f)−1 = 1 Φ( f ). f) Φ(f) is a bounded operator if and only if f ∈ L∞(K, ν). ∞ g) If (fn)n is a bounded sequence in L (K, ν) with fn → f ν-almost ev- p erywhere, then Φ(fn) → Φ(f) strongly on L (Ω).

Proof. a) Since Φ(f) = Mf◦a, this follows from Theorem 2.1.a). b) This is obvious. c) and d) follow from Theorem 2.1.e). e) follows from Theorem 2.1.b). f) follows from Theorem 2.1.c). g) It follows from the hypotheses that fn ◦ a → f ◦ a µ-almost everyhere. If x ∈ X = Lp(X) is arbitrary, then

Φ(fn)x = (fn ◦ a)x → (f ◦ a)x = Φ(f)x by Lebesgue’s theorem. In the following we shall base an abstract deﬁnition of “functional calculus” on some of the just listed properties. Before, let us note an interesting fact: the functional calculus Φ for Ma is determined by its restriction to those f ∈ L0(K, ν) that yield bounded operators. In fact, for f ∈ L0(K, ν) let

1 f h := and g := . 1 + |f|2 1 + |f|2

∞ g Then h, g ∈ L (K, ν). Moreover, f = h and Φ(h) is injective, hence

−1 1 g Φ(h) Φ(g) = Φ( h )Φ(g) = Φ( h ) = Φ(f), by Theorem 2.2.d) and since Φ(g) is a bounded operator.

2.4 Abstract Functional Calculus (I) — Deﬁnition

With the concrete model of multiplication operators at hand, we now turn to a more axiomatic treatment of functional calculi. Let F be an algebra with a unit element 1 and let X be a Banach space. A mapping Φ : F → C(X) 2.4 Abstract Functional Calculus (I) — Deﬁnition 31 from F to the set of closed operators on X is called a proto-calculus (or: F- proto-calculus) on X if the following axioms are satisﬁed (f, g ∈ F, λ ∈ C): (FC1) Φ(1) = I. (FC2) λΦ(f) ⊆ Φ(λf) and Φ(f) + Φ(g) ⊆ Φ(f + g). (FC3) Φ(f)Φ(g) ⊆ Φ(fg) and

dom(Φ(f)Φ(g)) = dom(Φ(g)) ∩ dom(Φ(fg)).

An element f ∈ F is called Φ-bounded if Φ(f) ∈ L(X). The set of Φ- bounded elements is denoted by

−1 FΦ := {f ∈ F : Φ(f) ∈ L(X)} = Φ (L(X)).

For each f ∈ F the set of its Φ-regularizers is

RegΦ(f) := {e ∈ F | e, ef ∈ FΦ}.

The deﬁnition of a regularizer here diﬀers from and is more general than the one given in our earlier work [1]. It has been noticed in [2] that such a relaxation of terminology is useful.

The following theorem summarizes basic properties of a proto-calculus.

Theorem 2.3. Let Φ : F → C(X) be a proto-calculus on a Banach space X. Then the following assertions hold (f, g ∈ F, λ ∈ C): a) If λ 6= 0 or Φ(f) ∈ L(X) then Φ(λf) = λΦ(f). b) If Φ(g) ∈ L(X) then

Φ(f) + Φ(g) = Φ(f + g) and Φ(f)Φ(g) = Φ(fg).

c) If fg = 1 then Φ(g) is injective and Φ(g)−1 ⊆ Φ(f). If, in addition fg = gf, then Φ(g)−1 = Φ(f).

d) The set FΦ of Φ-bounded elements is a unital subalgebra of F and

Φ : FΦ → L(X)

is an algebra homomorphism.

e) The set RegΦ(f) of Φ-regularizers of f is a left ideal in FΦ. Proof. a) One has

Φ(f) = Φ(λ−1λf) ⊇ λ−1Φ(λf) ⊇ λ−1λΦ(f) = Φ(f). 32 2 Unbounded Operators and Abstract Functional Calculus

Hence, all inclusions are equalities, and the assertion follows. b) By Axiom (FC2) and a)

Φ(f) = Φ(f + g − g) ⊇ Φ(f + g) + Φ(−g) = Φ(f + g) − Φ(g) ⊇ Φ(f) + Φ(g) − Φ(g) =! Φ(f).

Hence, all inclusions are equalities and the ﬁrst assertion in b) follows. For the second, note that by Axiom (FC3) Φ(f)Φ(g) ⊆ Φ(fg) with

dom(Φ(f)Φ(g)) = dom(Φ(g)) ∩ dom(Φ(fg)) = dom(Φ(fg)), hence we are done. c) If fg = 1 then Φ(f)Φ(g) ⊆ Φ(fg) = Φ(1) = I. Hence, Φ(g) is injective and Φ(f) ⊇ Φ(g)−1. If fg = gf here, then, by symmetry, one has that Φ(f) is injective, too, and Φ(g) ⊇ Φ(f)−1. This yields Φ(f) = Φ(g)−1 as desired. d) and e) follow directly from b).

Determination. Let Φ : F → C(X) be a proto-calculus on X and e, f ∈ F. Then, by Axiom (FC3), Φ(e)Φ(f) ⊆ Φ(ef). (2.1) If in addition Φ(e),Φ(ef) ∈ L(X), i.e., if e is a Φ-regularizer of f, then (2.1) simply means that

∀ x, y ∈ X : Φ(f)x = y ⇒ Φ(ef)x = Φ(e)y.

In very special situations, e.g. if e has a left inverse in F, the converse implication holds (Exercise 2.7). In that case, we say that e is Φ-determining for f. More generally, a subset M ⊆ RegΦ(f) is said to be Φ-determining for f ∈ F if one has

Φ(f)x = y ⇐⇒ ∀e ∈ M : Φ(ef)x = Φ(e)y (2.2) for all x, y ∈ X. (As explained, only the implication “⇐” is interesting here.)

Lemma 2.4. Let Φ be an F-proto-calculus on a Banach space X, let f ∈ F and let M ⊆ RegΦ(f) be Φ-determining for f. Then the following assertions hold: T a) e∈M ker(Φ(e)) = {0}. 0 0 b) Each set M with M ⊆ M ⊆ RegΦ(f) is also determining for f. c) If T ∈ L(X) commutes with all operators Φ(e) and Φ(ef) for e ∈ M, then T commutes with Φ(f).

Recall that the last statement simply means TΦ(f) ⊆ Φ(f)T . 2.4 Abstract Functional Calculus (I) — Deﬁnition 33

Proof. a) follows since Φ(f) is an operator and not just a linear relation. b) is clear. c) Suppose that Φ(f)x = y. Then, for each e ∈ M,

Φ(ef)T x = TΦ(ef)x = TΦ(e)y = Φ(e)T y.

Since M is Φ-determining for f, Φ(f)T x = T y. But this just means that TΦ(f) ⊆ Φ(f)T .

A proto-calculus Φ : F → C(X) is called a calculus (or: F-calculus) if in addition to (FC1)–(FC3) the following fourth axiom is satisﬁed:

(FC4) For each f ∈ F the set RegΦ(f) is Φ-determining for f.

The “Determination Axiom” (FC4) ensures that an F-proto-calculus Φ is determined in an algebraic way by its restriction to the set of Φ- bounded elements. If Φ(F) ⊆ L(X), i.e., if every f ∈ F is Φ-bounded, then (FC4) is trivially satisﬁed (why?).

Remarks 2.5. 1) The restriction of a proto-calculus Φ : F → C(X) to a unital subalgebra F 0 of F is again a proto-calculus, sloppily called a subcalculus. Even if the original proto-calculus satisﬁes (FC4), this need not be the case for the subcalculus. (It does if the subcalculus has the same set of Φ-bounded elements.) Similarly, the subcalculus may satisfy (FC4) while the original one does not.

2) Given a proto-calculus Φ : F → C(X), the set FΦ of Φ-bounded elements is a unital subalgebra and the restriction of Φ to FΦ a homomorphism of unital algebras. Let us call this the bounded subcalculus. As already mentioned, it trivially satisfies (FC4). 3) A proto-calculus Φ : F → C(X) such that Φ(F) ⊆ L(X) (equivalently: F = FΦ) is nothing else then a representation of the unital algebra F by bounded operators. So with our definition of a (proto-)calculus we generalize representation theory towards unbounded operators. However, Axiom (FC4) (if satis- fied) guarantees that the whole calculus is determined by the bounded subcalculus and we are confronted only with a “tame” kind of unbound- edness. 4) Universal regularizers other than the trivial one or universally determining sets need not exist, simply because the algebra F can be “too large”. Trivially, if F = FΦ, each f ∈ F is a universal regularizer. 34 2 Unbounded Operators and Abstract Functional Calculus

5) If the algebra F is not an algebra of functions, an F-calculus is—strictly speaking—not a functional calculus. However, although we shall try to be consistent with this distinction, a little sloppyness should be allowed here. After all, many commutative algebras are isomorphic to function algebras.

Example 2.6. Let Ω = (Ω, Σ, µ) be a semi-ﬁnite measure space and a : Ω → C a µ-essentially measurable function. Then, with ν = a∗µ, K := essran(a) and X = Lp(Ω), the mapping

0 Φ :L (K, ν) → C(X),Φ(f) := Mf◦a

0 0 is an L (K, ν)-calculus on X. The set of Φ-bounded elements is L (K, ν)Φ = L∞(K, ν). For f ∈ L0(K, ν) each function

2 −α 1 (1 + |f| ) , α ≥ 2 , is a Φ-determining Φ-regularizer for f.

2.5 Abstract Functional Calculus (II) — Generators

The deﬁnition of a (proto-)calculus given so far is, admittedly, very general. In many cases, the algebra F is actually a subalgebra of functions (or of equivalence classes of functions) on some subset D ⊆ C of the complex plane and contains the special function z := (z 7→ z). (E.g., in our multiplicator example, D = K = essran(a) and F = L0(K, ν).) In that case, the operator A := Φ(z) is a distinguished closed operator on X, and the F-calculus Φ can be called a functional calculus for A and A is called the generator of the functional calculus Φ. Moreover, the notation

f(A) := Φ(f) is employed.

Corollary 2.7. Let A be the generator of an F-proto-calculus on a Banach space X, where F is an algebra of functions on some subset D ⊆ C. Let λ ∈ C 1 such that λ−z ∈ F. Then λ − A is injective and the following assertions are equivalent: (i) λ ∈ ρ(A); 1 (ii)( λ−z )(A) ∈ L(X). 1 In this case, R(λ, A) = ( λ−z )(A). Proof. Let the given F-calculus be called Φ. Note that Φ(λ1) = λΦ(1) = λI ∈ L(X) and hence 2.5 Abstract Functional Calculus (II) — Generators 35

Φ(λ − z) = λI − A. 1 1 Since 1 = (λ − z) λ−z = λ−z (λ − z), by Theorem 2.3 the operator λ − A is −1 1 injective and (λ − A) = Φ( λ−z ). So λ ∈ ρ(A) if and only if this operator is bounded.

Example 2.8. Let Φ be the L0(K, ν)-calculus for some multiplication operator Ma as before. Then A = Φ(z) = Ma, so our terminology is consistent. For f ∈ L0(K, ν) one then has (by deﬁnition)

f(Ma) = Mf◦a.

E.g., since A is multiplication by a, eA is multiplication by ea. Note that here K = essran(a) = σ(A) and λ − A is injective precisely when (λ − z)−1 ∈ L0(K, ν). (Why?)

Corollary 2.7 hints at situations when one can identify a generator of a functional calculus Φ even in the case when the coordinate function z itself is 1 not contained in F, but some functions of the form λ−z are. This is treated in the following supplement.

Supplement: Linear relations as generators

Let Φ : F → C(X) be an F-calculus, where F is an algebra of functions deﬁned on some set D ⊆ C. We suppose in addition that the set

−1 ΛΦ := {λ ∈ C :(λ − z) ∈ F} is not empty. Then the following result yields a unique linear relation A on X such that

−1 −1 Φ (λ − z) = (λ − A) for all λ ∈ ΛΦ.

Theorem 2.9. Let Φ : F → C(X) be an F-calculus, where F is an algebra of functions on some set D ⊆ C. Then the linear relation 1 −1 λ − Φ λ − z is independent of λ ∈ ΛΦ.

1 1 Proof. Let λ, µ ∈ ΛΦ and deﬁne rλ(z) := λ−z and rµ(z) := µ−z . Then the resolvent identity (µ − λ)rλrµ = rλ − rµ holds. Now ﬁx eµ ∈ RegΦ(rµ) and eλ ∈ RegΦ(rλ). Then e := eµeλ is a regularizer for both rλ and rµ. Let, furthermore, x, y ∈ X. Note that 36 2 Unbounded Operators and Abstract Functional Calculus

−1 (x, y) ∈ λ − Φ(rλ) ⇐⇒ Φ(rλ)(λx − y) = x and likewise for µ instead of λ. We shall show that

Φ(rλ)(λx − y) = x =⇒ Φ(rµ)(µx − y) = x.

By symmetry, this will suﬃce for establishing the claim.

So abbreviate u := λx − y and suppose that Φ(rλ)u = x. Then Φ(eλrλ)u = Φ(eλ)x and hence

Φ(eλ)Φ(eµrµ)(µx − y) = Φ(erµ)((µ − λ)x + u)

= (µ − λ)Φ(eµrµ)Φ(eλ)x + Φ(erµ)u

= (µ − λ)Φ(eµrµ)Φ(eλrλ)u + Φ(erµ)u

= Φ(e(µ − λ)rµrλ + erµ)u

= Φ(erλ)u = Φ(eµ)Φ(eλrλ)u = Φ(eµ)Φ(eλ)x

= Φ(eλ)Φ(eµ)x.

Since eλ ranges over all of RegΦ(rλ),

Φ(eµrµ)(µx − y) = Φ(eµ)x.

But this holds for all eµ ∈ RegΦ(rµ), hence by Axiom (FC4) it follows that

Φ(rµ)(µx − y) = x.

Suppose that Φ is an F-calculus as above, and such that ΛΦ 6= ∅. Then we call the linear relation

−1 −1 AΦ := λ − Φ((λ − z) ) (λ ∈ ΛΦ) the generator of the calculus Φ.

Exercises

2.1. Let X and Y be Banach spaces and let A : X ⊇ dom(A) → Y be a linear operator. a) Prove that the following assertions are equivalent: (i) A is closed.

(ii) Whenever dom(A) 3 xn → x ∈ X and Axn → y ∈ Y then x ∈ dom(A) and Ax = y.

(iii) The space (dom(A), k · kA) is a Banach space. 2.5 Abstract Functional Calculus (II) — Generators 37

b) Prove that each two of the following three assertions together imply the third one: (i) A is continuous for the norm of X, i.e., there is c ≥ 0 such that kAxk ≤ ckxk for all x ∈ dom(A). (ii) A is closed. (iii) dom(A) is a closed subspace of X. 2.2. Let A, B, C be linear operators on a Banach space X. a)( A + B)C = AC + BC. b) C(A+B) ⊇ CA+CB, with equality if, for instance, ran(A) ⊆ dom(C). c) If A is closed and B is bounded, then AB is closed. d) If B is closed and A is invertible, then AB is closed. 2.3. Let A and B be linear operators on a Banach space X, T ∈ L(X) and λ ∈ ρ(A) ∩ ρ(B). Show that the following assertions are equivalent: (i) For all x, y ∈ X, if Ax = y then BT x = T y; (ii) TA ⊆ BT ; (iii) TR(λ, A) = R(λ, B)T for one/all λ ∈ ρ(A) ∩ ρ(B). 2.4 (Spectral Mapping Theorem for the Inverse). Let A be linear relation on X and let λ ∈ K \{0}. Show that

1 −1−1 2 −1 λ − A = λ − λ (λ − A) . Conclude that

−1 1 ρ(A) \{0} → ρ(A ) \{0}, λ 7→ λ is a bijection. 2.5 (Multiplication Operator on Continuous Functions). Let (Ω, d) be a metric space. For a continuous function a ∈ C(Ω) let Ma be the multiplication operator deﬁned on Cb(Ω) by

Maf := af for f ∈ dom(Ma) := {f ∈ Cb(Ω): af ∈ Cb(Ω)}.

a) Show that Ma is a closed operator but need not be densely deﬁned.

b) Show that Ma is bounded if and only if a is bounded, and kMak = kak∞ in this case. c) Show the analogue of e) and f) from Theorem 2.1.

d) Show that σ(Ma) = σa(Ma) = a(Ω). e) Formulate and prove analogues of h) and i) of Theorem 2.1. 38 2 Unbounded Operators and Abstract Functional Calculus

2.6. Let Ω = (Ω, Σ, µ) be a measure space and a : Ω → C measurable. Show that a ∈ essran(a) µ-almost everywhere.

2.7. Let Φ be an F-proto-calculus on a Banach space X, let e, f ∈ F such that Φ(e),Φ(ef) ∈ L(X). Suppose that e has a left inverse e0 in F, i.e., e0e = 1. Show that Φ(f) = Φ(e)−1Φ(ef) and

Φ(f)x = y ⇐⇒ Φ(ef)x = Φ(e)y for x, y ∈ X.

2.8. Let Φ be the L0(K, ν)-calculus on X = Lp(Ω) for some multiplication operator Ma, a : Ω → C. Examine, under which assumptions this calculus has a nontrivial universal regularizer. [A characterization is possible, but not straightforward.]

2.9. Let, as in Exercise 2.5, Ω be a metric space and a ∈ C(Ω). Let D := a(Ω) ⊆ C. For a continuous function f : D → C deﬁne

Φ(f) := Mf◦a.

Show that Φ : C(D) → C(Cb(Ω)) is a functional calculus for Ma, and each f ∈ C(D) has a determining regularizer. Can you think of an example showing that in general the given functional calculus is not “maximal” (whatever this means precisely)?

2.10. Let Φ : F → C(X) be an F-(proto)-calculus. a) Suppose that F is commutative. Show that N := {f ∈ F : Φ(f) = 0} is an ideal of F and that by

Φ(f + N ) := Φ(f)

a F/N -(proto)-calculus is deﬁned. b) Let F 0 be another unital algebra and Ψ : F 0 → F a unital algebra homomorphism. Show that Φ ◦ Ψ : F 0 → C(X) is a F 0-proto-calculus. If Φ is a calculus and Ψ is surjective, then Φ ◦ Ψ is a calculus as well.

References

[1] M. Haase. The Functional Calculus for Sectorial Operators. Vol. 169. Operator Theory: Advances and Applications. Basel: Birkh¨auserVerlag, 2006, pp. xiv+392. [2] M. Haase. “On abstract functional calculus extensions”. In: T¨ubinger Berichte zur Funktionalanalysis (2008).