PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 134, Number 8, Pages 2383–2392 S 0002-9939(06)08258-X Article electronically published on March 21, 2006 ON EXTENDED EIGENVALUES AND EXTENDED EIGENVECTORS OF SOME OPERATOR CLASSES M. T. KARAEV (Communicated by N. Tomczak-Jaegermann) Abstract. We give a complete description of the set of extended eigenvectors x of the Volterra integration operator V, V f(x)= f(t)dt,onL2 [0, 1], which 0 strengthens the result of a paper by Biswas, Lambert, and Petrovic (2002). We also introduce the concept of a well splitting operator and study its extended eigenvalues and extended eigenvectors. 1. Introduction and preliminaries Let E be a Banach space, and denote by B(E) the algebra of all bounded linear operators on E.IfA is an operator in B(E)andλ is a complex number, then following Biswas, Lambert and Petrovic [1] we say that a complex number λ is an extended eigenvalue of A if there exists a nonzero operator X in B(E)such that AX=λXA ;suchanoperatorX is called extended eigenvector corresponding to λ. The set of all extended eigenvalues of A will be called the extended point ext ∈ ext spectrum, and it will be denoted as σp (A). For a given λ σp (A) we define {A} as the set of all λ-extended eigenvectors for A ({A} is {A} , the commu- λ 1 { } tant of A). The set of all extended eigenvectors for A will be denoted as A ext, { } { } i.e., A ext = A λ. The basic facts about the extended eigenvalues and ∈ ext λ σp (A) extended eigenvectors of operators can be found in [1]–[3]. In what follows, V will denote the simple Volterra integration operator on L2 [0, 1] defined as x (Vf)(x)= f(t)dt. 0 It was established in [1] that the set of extended eigenvalues of the Volterra integra- tion operator V is precisely the set (0, ∞). Moreover, it was shown in [1], Theorem 6, that for each such extended eigenvalue λ, the appropriate extended eigenvector can be found in the class of integral operators. In other words, for each λ>0, the equation XV = λV X Received by the editors March 3, 2005 and, in revised from, March 14, 2005. 2000 Mathematics Subject Classification. Primary 47A15. Key words and phrases. Extended eigenvalue, extended eigenvector, Volterra integration operator. c 2006 American Mathematical Society 2383 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2384 M. T. KARAEV has a nonzero integral operator as a solution. In this paper, we give a complete { } description of the set V ext of extended eigenvectors of the integration operator V on L2 [0, 1] (see Theorem 1 in Section 2) which strengthens Theorem 6 of the paper [1]. In Section 3 we study the set of extended eigenvalues and extended eigenvectors of so-called well splitting operators on a Banach space E. In Section 4 we give an application of Deddens algebras and Shulman subspaces to the “extended” spectral theory. 2. Extended eigenvectors for V Let ϕ :[0, 1] → [0, 1] be a measurable function. A composition operator Cϕ is defined as Cϕf(x)=f (ϕ(x)). The composition operator Cλx,whereλ ∈ [0, 1], will 2 be denoted simply as Cλ. The so-called Duhamel operator Df on L [0, 1] is defined as x def def d (1) D g =(f g)(x) = f(x − t)g(t)dt. f dx 0 { } The following result describes the set V ext of Volterra integration operator V on L2 [0, 1]. Theorem 1. Let λ ∈ (0, ∞) and let X ∈B L2 [0, 1] be a nonzero operator. Then: (i) if λ ≤ 1,thenXV = λV X if and only if X = DX1Cλ, that is, x d (Xf)(x)= (X1)(x − t)f(λt)dt, f ∈ L2 [0, 1] ; dx 0 (ii) if λ>1,thenXV = λV X if and only if XC1/λ = DX1. Proof. (i) Let XV = λV X.Then XV n = λnV nX for any n ≥ 1, that is, XV nf = λnV nXf for all f ∈ L2 [0, 1]. In particular, XV n1 = λnV nX1, that is, xn−1 xn−1 X ∗ 1 = λn ∗ X1, (n − 1)! (n − 1)! or xn xn−1 X = λn ∗ X1,n≥ 1. n! (n − 1)! ∗−multiplying the last equality by 1 we obtain that xn xn−1 1 ∗ X = λn 1 ∗ ∗ X1 , n! (n − 1)! that is, xn (λx)n 1 ∗ X = ∗ X1 , n! n! for any n ≥ 1. Hence 1 ∗ Xp(x)=p (λx) ∗ X1 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use EXTENDED EIGENVALUES AND EIGENVECTORS OF OPERATOR CLASSES 2385 for all polynomials p.SinceL2 [0, 1] is a Banach algebra with respect to the convo- lution product ∗, it follows from the last equality that 1 ∗ Xf(x)=f (λx) ∗ X1 for all f ∈ L2 [0, 1]. Since Vf(x)=1 ∗ f for every f ∈ L2 [0, 1], the last equality means that VXf(x)=KX1Cλf(x), 2 def where KX1 denotes the usual convolution operator on L [0, 1], KX1g(x) = X1 ∗ x − ∈ 2 d g = (X1)(x t) g(t)dt, g L [0, 1]. From this, by applying operator dx we 0 obtain that d (Xf)(x)= K C f(x), dx X1 λ that is, x d (2) (Xf)(x)= (X1)(x − t) f(λt)dt , dx 0 or Xf = DX1Cλf 2 for all f ∈ L [0, 1], which means that X = DX1Cλ. Now it suffices to show that if 2 X = DX1Cλ,thenXV = λV X.Infact,forallf ∈ L [0, 1] we have XV f(x)=DX1Cλf(x)=DX1 (Vf)(λx) = X1 (Vf)(λx)=X1 (λx f (λx)) = λx (X1 f (λx)) = λx DX1Cλf(x) = λ (x DX1Cλf(x)) = λV DX1Cλf(x) = λV Xf(x), which completes the proof of (i). 1 n 1 n (ii) Let XV = λV X. Clearly VX = λ XV , which implies that V X = X λ V for every n ≥ 1. The same arguments as in the proof of (i) yield the equality x Xf = X1 f, f ∈ L2 [0, 1] , λ where is the Duhamel product defined by (1). Conversely, if XC1/λ = DX1,then since Vf(x)=x f and DX1V = V DX1, for all polynomials p we have λV Xp(x)=λV XC1/λCλp(x) = λV DX1Cλp(x)=λV DX1p (λx) = λDX1Vp(λx)=λXC1/λVp(λx) = λXC1/λ (x p(λx)) = XC1/λ (λx p(λx)) = XC1/λ (Vp)(λx)=XV p(x), which completes the proof of (ii) (because the set of polynomials is dense in L2[0, 1]). Theorem 1 is proved. The following result was proved by a different method in [1], Theorem 7. Corollary 2. The composition operator Cϕ satisfies the equation CϕV = λV Cϕ, where λ ∈ (0, 1],ifandonlyifϕ(x)=λx , x ∈ [0, 1]. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2386 M. T. KARAEV Proof. It is obvious that Cϕ1 = 1. Then according to part (i) of Theorem 1 we have that CϕV = λV Cϕ if and only if x d C f(x)= f(λt)dt, ϕ dx 0 that is, Cϕf(x)=f(λx)=Cλf(x) 2 for all f ∈ L [0, 1], i.e., Cϕ = Cλ ,orϕ(x)=λx. This completes the proof. 3. Well splitting operators Let X be a separable Banach space and A ∈B(X). An operator A is called a splitting operator in X if for every x ∈ X there exists a linear densely defined operator Bx (generally nonbounded) such that n (3) A x = Bxyn { } for every n, n =0, 1, 2, ..., and for some complete system yn n≥0 of the space X. We say that the splitting operator A is well splitting if for every x ∈ X the corresponding operators Bx in (3) are bounded in X. It is immediate from the last definition that a well splitting operator is cyclic if for some x ∈ X an operator Bx has dense range in X. It is easy to see that the concept of splitting operator is a generalization of the so-called basis operator introduced by Nikolski [4]. (n) (n) Example 1. If X denotes any of the spaces C [0, 1] and Wp [0, 1] (Sobolev space), 1 ≤ p<+∞, then it is not difficult to show that the Volterra integration operator V acting in X is a well splitting operator. Indeed, let us denote by the Duhamel product defined as x d (4) (f g)(x)= f(x − t)g(t)dt dx 0 for all f,g ∈ X. Simple calculations show that X is a Banach algebra with respect to the Duhamel product .Inparticular,foranyf ∈ X the “Duhamel operator” D D def f , f g = f g, is a bounded operator in X. Now it follows immediately from (4) that xn V nf = f , n ≥ 0, n! that is, xn (5) V nf = D , n ≥ 0, f n! ∈ xn for all f X. By Weierstrass’s approximation theorem n! n≥0 is a complete system in X, and hence, formula (5) means that V is a well splitting operator in X. { } In what follows, for every complete system yn n≥0 in X we will denote by the symbol Λ{yn} the set of all complex numbers λ for which the diagonal operator def n D{λ}, D{λ}yn = λ yn, n ≥ 0, is bounded in X. The following result gives, in particular, more general examples of well splitting operators.