Continuity of spectrum and approximate point spectrum on operator matrices

Let X and Y be given Banach spaces. For A∈B(X), B∈B(Y) and C∈B(Y,X), let MC be the operator defined on X⊕Y by . In this paper we give conditions for continuity of τ at MC through continuity of τ at A and B, where τ can be equal to the spectrum or approximate point spectrum.     

Browder-Weyl theorems, tensor products and multiplications

A Banach space operator T∈B(X) satisfies Browder's theorem if the complement of the Weyl spectrum σw(T) of T in σ(T) equals the set of Riesz points of T; T is polaroid if the isolated points of σ(T) are poles (no restriction on rank) of the resolvent of T. Let Φ(T) denote the set of Fredholm points of T. Browder's theorem transfers from A,B∈B(X) to S=LARB (resp., S=A⊗B) if and only if A and B^∗ (resp., A and B) have SVEP at points μ∈Φ(A) and ν∈Φ(B) for which λ=μν∉σw(S). If A and B are finitely polaroid, then the polaroid property transfers from A∈B(X) and B∈B(Y) to LARB; again, restricting ourselves to the completion of X⊗Y in the projective topology, if A and B are finitely polaroid, then the polaroid property transfers from A∈B(X) and B∈B(Y) to A⊗B.     

Essential point spectra of operator matrices trough local spectral theory

For A∈B(X), B∈B(Y) and C∈B(Y,X), let MC be the operator defined on X⊕Y by . In this paper, we study defect set (Σ(A)∪Σ(B))?Σ(MC), where Σ is the Browder spectrum, the essential approximate point spectrum and Browder essential approximate point spectrum. We then give application for Weyl's and Browder's theorems.     

Maps preserving peripheral spectrum of Jordan semi-triple products of operators

Let A and B be (not necessarily unital or closed) standard operator algebras on complex Banach spaces X and Y, respectively. For a bounded linear operator A on X, the peripheral spectrum σ_π(A) of A is the set σ_π(A)={z∈σ(A):|z|=max_ω∈σ(A)|ω|}, where σ(A) denotes the spectrum of A. Assume that Φ:A→B is a map the range of which contains all operators of rank at most two. It is shown that the map Φ satisfies the condition that σ_π(BAB)=σ_π(Φ(B)Φ(A)Φ(B)) for all A,B∈A if and only if there exists a scalar λ∈C with λ³=1 and either there exists an invertible operator T∈B(X,Y) such that Φ(A)=λTAT^-1 for every A∈A; or there exists an invertible operator T∈B(X^∗,Y) such that Φ(A)=λTA^∗T^-1 for every A∈A. If X=H and Y=K are complex Hilbert spaces, the maps preserving the peripheral spectrum of the Jordan skew semi-triple product BA^∗B are also characterized. Such maps are of the form A?UAU^∗ or A?UA^tU^∗, where U∈B(H,K) is a unitary operator, A^t denotes the transpose of A in an arbitrary but fixed orthonormal basis of H.     

On operator valued sequences of multipliers and R-boundedness

In recent papers (cf. [J.L. Arregui, O. Blasco, (p,q)-Summing sequences, J. Math. Anal. Appl. 274 (2002) 812-827; J.L. Arregui, O. Blasco, (p,q)-Summing sequences of operators, Quaest. Math. 26 (2003) 441-452; S. Aywa, J.H. Fourie, On summing multipliers and applications, J. Math. Anal. Appl. 253 (2001) 166-186; J.H. Fourie, I. Röntgen, Banach space sequences and projective tensor products, J. Math. Anal. Appl. 277 (2) (2003) 629-644]) the concept of (p,q)-summing multiplier was considered in both general and special context. It has been shown that some geometric properties of Banach spaces and some classical theorems can be described using spaces of (p,q)-summing multipliers. The present paper is a continuation of this study, whereby multiplier spaces for some classical Banach spaces are considered. The scope of this research is also broadened, by studying other classes of summing multipliers. Let E(X) and F(Y) be two Banach spaces whose elements are sequences of vectors in X and Y, respectively, and which contain the spaces c₀₀(X) and c₀₀(Y) of all X-valued and Y-valued sequences which are eventually zero, respectively. Generally spoken, a sequence of bounded linear operators (un)⊂L(X,Y) is called a multiplier sequence from E(X) to F(Y) if the linear operator from c₀₀(X) into c₀₀(Y) which maps (xi)∈c₀₀(X) onto (unxn)∈c₀₀(Y) is bounded with respect to the norms on E(X) and F(Y), respectively. Several cases where E(X) and F(Y) are different (classical) spaces of sequences, including, for instance, the spaces Rad(X) of almost unconditionally summable sequences in X, are considered. Several examples, properties and relations among spaces of summing multipliers are discussed. Important concepts like R-bounded, semi-R-bounded and weak-R-bounded from recent papers are also considered in this context.     

Hereditarily polaroid operators, SVEP and Weyl's theorem

A Banach space operator T∈B(X) is hereditarily polaroid, T∈HP, if every part of T is polaroid. HP operators have SVEP. It is proved that if T∈B(X) has SVEP and R∈B(X) is a Riesz operator which commutes with T, then T+R satisfies generalized a-Browder's theorem. If, in particular, R is a quasi-nilpotent operator Q, then both T+Q and T^∗+Q^∗ satisfy generalized a-Browder's theorem; furthermore, if Q is injective, then also T+Q satisfies Weyl's theorem. If A∈B(X) is an algebraic operator which commutes with the polynomially HP operator T, then T+N is polaroid and has SVEP, f(T+N) satisfies generalized Weyl's theorem for every function f which is analytic on a neighbourhood of σ(T+N), and f^∗(T+N) satisfies generalized a-Weyl's theorem for every function f which is analytic on, and constant on no component of, a neighbourhood of σ(T+N).     

Functions of operators under perturbations of class Sp

This is a continuation of our paper [2]. We prove that for functions f in the Hölder class Λα(R) and 1<p<∞, the operator f(A)−f(B) belongs to Sp/α, whenever A and B are self-adjoint operators with A−B∈Sp. We also obtain sharp estimates for the Schatten-von Neumann norms ‖f(A)−f(B)Sp/α_‖ in terms of ‖A−BSp_‖ and establish similar results for other operator ideals. We also estimate Schatten-von Neumann norms of higher order differences . We prove that analogous results hold for functions on the unit circle and unitary operators and for analytic functions in the unit disk and contractions. Then we find necessary conditions on f for f(A)−f(B) to belong to Sq under the assumption that A−B∈Sp. We also obtain Schatten-von Neumann estimates for quasicommutators f(A)R−Rf(B), and introduce a spectral shift function and find a trace formula for operators of the form f(A−K)−2f(A)+f(A+K).     

Perturbation of spectra of operator matrices and local spectral theory

For A∈L(X), B∈L(Y) and C∈L(Y,X) we denote by MC the operator defined on X⊕Y by . In this article, we study defect set DΣ=(Σ(A)∪Σ(B))?Σ(MC) for different spectra including the spectrum, the essential spectrum, Weyl spectrum and the approximate point spectrum. We then apply the obtained results to the stability of such spectra (DΣ=∅) and the classes of operators C for which stability holds of MC using local spectral theory.     

Banach spaces of operators that are complemented in their biduals

Let [A, a] be a normed operator ideal. We say that [A, a] is boundedly weak*-closed if the following property holds: for all Banach spaces X and Y, if T: X → Y** is an operator such that there exists a bounded net (T _i ) _i∈I in A(X, Y) satisfying lim _i 〈y*, T _i x y*〉 for every x ∈ X and y* ∈ Y*, then T belongs to A(X, Y**). Our main result proves that, when [A, a] is a normed operator ideal with that property, A(X, Y) is complemented in its bidual if and only if there exists a continuous projection from Y** onto Y, regardless of the Banach space X. We also have proved that maximal normed operator ideals are boundedly weak*-closed but, in general, both concepts are different.      

Weyl's theorem for algebraically totally hereditarily normaloid operators

A Banach space operator T∈B(X) is said to be totally hereditarily normaloid, T∈THN, if every part of T is normaloid and every invertible part of T has a normaloid inverse. The operator T is said to be an H(q) operator for some integer q?1, T∈H(q), if the quasi-nilpotent part H₀(T−λ)=(T−λ)^−q(0) for every complex number λ. It is proved that if T is algebraically H(q), or T is algebraically THN and X is separable, then f(T) satisfies Weyl's theorem for every function f analytic in an open neighborhood of σ(T), and T^∗ satisfies a-Weyl's theorem. If also T^∗ has the single valued extension property, then f(T) satisfies a-Weyl's theorem for every analytic function f which is non-constant on the connected components of the open neighborhood of σ(T) on which it is defined.     

The best generalized inverse of the linear operator in normed linear space

Let X,Y be normed linear spaces, T∈L(X,Y) be a bounded linear operator from X to Y. One wants to solve the linear problem Ax=y for x (given y∈Y), as well as one can. When A is invertible, the unique solution is x=A^-1y. If this is not the case, one seeks an approximate solution of the form x=By, where B is an operator from Y to X. Such B is called a generalised inverse of A. Unfortunately, in general normed linear spaces, such an approximate solution depends nonlinearly on y. We introduce the concept of bounded quasi-linear generalised inverse T^h of T, which contains the single-valued metric generalised inverse T^M and the continuous linear projector generalised inverse T⁺. If X and Y are reflexive, we prove that the set of all bounded quasi-linear generalised inverses of T, denoted by G_H(T), is not empty In the normed linear space of all bounded homogeneous operators, the best bounded quasi-linear generalised inverse T^h of T is just the Moore-Penrose metric generalised inverse T^M. In the case, X and Y are finite dimension spaces Rⁿ and R^m, respectively, the results deduce the main result by G.R. Goldstein and J.A. Goldstein in 2000.     

Rank-1 perturbations of cosine functions and semigroups

Let A be the generator of a cosine function on a Banach space X. In many cases, for example if X is a UMD-space, A+B generates a cosine function for each B∈L(D((ω−A)^1/2),X). If A is unbounded and , then we show that there exists a rank-1 operator B∈L(D(γ(ω−A)),X) such that A+B does not generate a cosine function. The proof depends on a modification of a Baire argument due to Desch and Schappacher. It also allows us to prove the following. If A+B generates a distribution semigroup for each operator B∈L(D(A),X) of rank-1, then A generates a holomorphic C₀-semigroup. If A+B generates a C₀-semigroup for each operator B∈L(D(γ(ω−A)),X) of rank-1 where 0<γ<1, then the semigroup T generated by A is differentiable and ‖T^′(t)‖=O(t−α) as t↓0 for any α>1/γ. This is an approximate converse of a perturbation theorem for this class of semigroups.     

Continuity of the spectrum on a class of upper triangular operator matrices

Let B(H) denote the algebra of operators on an infinite dimensional complex Hilbert space H, and let A^○∈B(K) denote the Berberian extension of an operator A∈B(H). It is proved that the set theoretic function σ, the spectrum, is continuous on the set C(i)⊂B(Hi) of operators A for which σ(A)={0} implies A is nilpotent (possibly, the 0 operator) and at every non-zero λ∈σp(A^○) for some operators X and B such that λ∉σp(B) and σ(A^○)={λ}∪σ(B). If CS(m) denotes the set of upper triangular operator matrices , where Aii∈C(i) and Aii has SVEP for all 1?i?m, then σ is continuous on CS(m). It is observed that a considerably large number of the more commonly considered classes of Hilbert space operators constitute sets C(i) and have SVEP.     

On the essential commutant of analytic Toeplitz operators associated with spherical isometries

Let T∈Bn(H) be an essentially normal spherical isometry with empty point spectrum on a separable complex Hilbert space H, and let AT⊂B(H) be the unital dual operator algebra generated by T. In this note we show that every operator S∈B(H) in the essential commutant of AT has the form S=X+K with a T-Toeplitz operator X and a compact operator K. Our proof actually covers a larger class of subnormal operator tuples, called A-isometries, which includes for example the tuple T=(Mz₁,…,Mzn)∈B(H²n(σ)) consisting of the multiplication operators with the coordinate functions on the Hardy space H²(σ) associated with the normalized surface measure σ on the boundary ∂D of a strictly pseudoconvex domain D⊂Cn. As an application we determine the essential commutant of the set of all analytic Toeplitz operators on H²(σ) and thus extend results proved by Davidson (1977) [6] for the unit disc and Ding and Sun (1997) [11] for the unit ball.     

On multiplicative perturbation of integral resolvent families

In this paper we study multiplicative perturbations for the generator of a strongly continuous integral resolvent family of bounded linear operators defined on a Banach space X. Assuming that a(t) is a creep function which satisfies a(⁰⁺)>0, we prove that if (A,a) generates an integral resolvent, then (A(I+B),a) also generates an integral resolvent for all B∈B(X,Z), where Z belongs to a class of admissible Banach spaces. In special instances of a(t) the space Z is proved to be characterized by an extended class of Favard spaces.     

An elementary operator with log-hyponormal, p-hyponormal entries

A Hilbert space operator A∈B(H) is p-hyponormal, A∈(p-H), if |A^∗|^2p?|A|^2p; an invertible operator A∈B(H) is log-hyponormal, A∈(?-H), if log(TT^∗)?log(T^∗T). Let d_AB=δ_AB or ?_AB, where δ_AB∈B(B(H)) is the generalised derivation δ_AB(X)=AX-XB and ?_AB∈B(B(H)) is the elementary operator ?_AB(X)=AXB-X. It is proved that if A,B^∗∈(?-H)∪(p-H), then, for all complex λ, , the ascent of (d_AB-λ)?1, and d_AB satisfies the range-kernel orthogonality inequality ‖X‖?‖X-(d_AB-λ)Y‖ for all X∈(d_AB-λ)^-1(0) and Y∈B(H). Furthermore, isolated points of σ(d_AB) are simple poles of the resolvent of d_AB. A version of the elementary operator E(X)=A₁XA₂-B₁XB₂ and perturbations of d_AB by quasi-nilpotent operators are considered, and Weyl’s theorem is proved for d_AB.     

Lower quasicontinuity, joint continuity and related concepts

Let X and Y be topological spaces, let Z be a metric space, and let f:X×Y→Z be a mapping. It is shown that when Y has a countable base B, then under a rather general condition on the set-valued mappings X∋x→fx(B)∈Z², B∈B, there is a residual set R⊂X such that for every (a,b)∈R×Y, f is jointly continuous at (a,b) if (and only if) fa:Y→Z is continuous at b. Several new results are also established when the notion of continuity is replaced by that of quasicontinuity or by that of cliquishness. Our approach allows us to unify and improve various results from the literature.     

Generalizations of weakly peripherally multiplicative maps between uniform algebras

Let A and B be uniform algebras on first-countable, compact Hausdorff spaces X and Y, respectively. For f∈A, the peripheral spectrum of f, denoted by σπ(f)={λ∈σ(f):|λ|=‖f‖}, is the set of spectral values of maximum modulus. A map T:A→B is weakly peripherally multiplicative if σπ(T(f)T(g))∩σπ(fg)≠∅ for all f,g∈A. We show that if T is a surjective, weakly peripherally multiplicative map, then T is a weighted composition operator, extending earlier results. Furthermore, if T₁,T₂:A→B are surjective mappings that satisfy σπ(T₁(f)T₂(g))∩σπ(fg)≠∅ for all f,g∈A, then T₁(f)T₂(1)=T₁(1)T₂(f) for all f∈A, and the map f?T₁(f)T₂(1) is an isometric algebra isomorphism.     

On additive polynomials and certain maximal curves

We show that a maximal curve over F_q² given by an equation A(X)=F(Y), where A(X)∈F_q²[X] is additive and separable and where F(Y)∈F_q²[Y] has degree m prime to the characteristic p, is such that all roots of A(X) belong to F_q². In the particular case where F(Y)=Y^m, we show that the degree m is a divisor of q+1.