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.     

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.     

Towards a maximal extension of Pe?czyński's decomposition method in Banach spaces

Suppose that X, Y, A and B are Banach spaces such that X is isomorphic to Y⊕A and Y is isomorphic to X⊕B. Are X and Y necessarily isomorphic? In this generality, the answer is no, as proved by W.T. Gowers in 1996. In the present paper, we provide a very simple necessary and sufficient condition on the 10-tuples (k,l,m,n,p,q,r,s,u,v) in N with p+q+u?3, r+s+v?3, uv?1, (p,q)≠(0,0), (r,s)≠(0,0) and u=1 or v=1 or (p,q)=(1,0) or (r,s)=(0,1), which guarantees that X is isomorphic to Y whenever these Banach spaces satisfy

     

Points d’évaluation pour les opérateurs cycliques ayant la propriété de Bishop (β)

Let T be a linear bounded cyclic operator in a separable complex Hilbert space H. Let B(T) and B_a(T) denote, respectively, the set of bounded point evaluation and the set of analytic point evaluation of T. We show that if T has the Bishop property (β), then B_a(T)=B(T)?σ_ap(T), where σ_ap(T) is the approximate spectrum of T. In the particular case when T is an operator of multiplication by z in a Hardy space this was proved by Trent (Pacific J. Math. 80 (1979) 279). On the other hand, using the generalized and the local spectral theory we obtain sufficient conditions on B_a(T) under which the spectrum of T and the local spectrum of T at any y≠0 in H coincide. At the end results involving the spectral picture of quasi-similar cyclic operators are given.     

M(r, s)-ideals of compact operators

We study the position of compact operators in the space of all continuous linear operators and its subspaces in terms of ideals. One of our main results states that for Banach spaces X and Y the subspace of all compact operators K (X, Y) is an M(r ₁ r ₂, s ₁ s ₂)-ideal in the space of all continuous linear operators L(X, Y) whenever K (X,X) and K (Y, Y) are M(r ₁, s ₁)- and M(r ₂, s ₂)-ideals in L(X,X) and L(Y, Y), respectively, with r ₁ + s ₁/2 > 1 and r ₂ +s ₂/2 > 1. We also prove that the M(r, s)-ideal K (X, Y ) in L(X, Y ) is separably determined. Among others, our results complete and improve some well-known results on M-ideals.     

The strongest intrinsic meaning of sequential-evaluation convergence

For classical Banach sequence spaces c₀(X), l_∞(X) and lp(X) (0<p<+∞) we have found the strongest intrinsical meanings of their β-duals, and two basic convergence results are established in the β-duals.     

Multiplicative versions of Klyachko’s theorem in finite factors

Let A and B be invertible positive elements in a II₁-factor A, and let μ_s(·) be the singular number on A. We prove that

exp_∫Klogμ_s(AB)ds?exp_∫Ilogμ_s(A)ds·exp_∫Jlogμ_s(B)ds,     

The intersection of left (right) spectra of 2 × 2 upper triangular operator matrices

When A∈B(H) and B∈B(K) are given, we denote by M_C the operator matrix acting on the infinite-dimensional separable Hilbert space H⊕K of the form In this paper, for given A and B, the sets and ?_C∈Inv(K,H)σ_l(M_C) are determined, where σ_l(T),B_l(K,H) and Inv(K,H) denote, respectively, the left spectrum of an operator T, the set of all the left invertible operators and the set of all the invertible operators from K into H.     

Continuous spectrum, point spectrum and residual spectrum of operator matrices

Let M_C denote a 2 × 2 upper triangular operator matrix of the form , which is acting on the sum of Banach spaces X⊕Y or Hilbert spaces H⊕K. In this paper, the sets and ?_C∈B(K,H)σ_r(M_C) are, respectively, characterized completely, where σ_c(·) denotes the continuous spectrum, σ_p(·) denotes the point spectrum and σ_r(·) denotes the residual spectrum. Moreover, some corresponding counterexamples are given.     

Simultaneous solutions of Sylvester equations and idempotent matrices separating the joint spectrum

We investigate simultaneous solutions of the matrix Sylvester equations A_iX-XB_i=C_i,i=1,2,…,k, where {A₁,…,A_k} and {B₁,…,B_k} are k-tuples of commuting matrices of order m×m and p×p, respectively. We show that the matrix Sylvester equations have a unique solution X for every compatible k-tuple of m×p matrices {C₁,…,C_k} if and only if the joint spectra σ(A₁,…,A_k) and σ(B₁,…,B_k) are disjoint. We discuss the connection between the simultaneous solutions of Sylvester equations and related questions about idempotent matrices separating disjoint subsets of the joint spectrum, spectral mapping for the differences of commuting k-tuples, and a characterization of the joint spectrum via simultaneous solutions of systems of linear equations.     

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.     

Two inequalities for the Perron Root

If A,B are irreducible, nonnegative n×n matrices with a common right eigenvector and a common left eigenvector corresponding to their respective spectral radii r(A), r(B), then it is shown that for any tϵ[0, 1], r(tA+(1−t)B^t)⩾tr(A)+ (1−t)r(B), where B^t is the transpose of B. Another inequality is proved that involves r(A) and r(Σ_lD^lAE^l), where A is a nonnegative, irreducible matrix and D^l, E^l are positive definite diagonal matrices. These inequalities generalize previous results due to Levinger and due to Friedland and Karlin.     

On orthogonal systems of matrix algebras

In this work it is shown that certain interesting types of orthogonal system of subalgebras (whose existence cannot be ruled out by the trivial necessary conditions) cannot exist. In particular, it is proved that there is no orthogonal decomposition of M_n(C)⊗M_n(C)≡M_n²(C) into a number of maximal abelian subalgebras and factors isomorphic to M_n(C) in which the number of factors would be 1 or 3.In addition, some new tools are introduced, too: for example, a quantity c(A,B), which measures “how close” the subalgebras A,B⊂M_n(C) are to being orthogonal. It is shown that in the main cases of interest, c(A^′,B^′) - where A^′ and B^′ are the commutants of A and B, respectively - can be determined by c(A,B) and the dimensions of A and B. The corresponding formula is used to find some further obstructions regarding orthogonal systems.     

A Grothendieck-type theorem for the space of totally measurable functions

Let Σ be a σ-algebra of subsets of a non-empty set Ω. Let X be a real Banach space and let X^* stand for the Banach dual of X. Let B(Σ, X) be the Banach space of Σ-totally measurable functions f: Ω → X, and let B(Σ, X)^* and B(Σ, X)^** denote the Banach dual and the Banach bidual of B(Σ, X) respectively. Let bvca(Σ, X^*) denote the Banach space of all countably additive vector measures ν: Σ → X^* of bounded variation. We prove a form of generalized Vitali-Hahn-Saks theorem saying that relative σ(bvca(Σ, X^*), B(Σ, X))-sequential compactness in bvca(Σ, X^*) implies uniform countable additivity. We derive that if X reflexive, then every relatively σ(B(Σ, X)^*, B(Σ, X))-sequentially compact subset of B(Σ, X)_c^~ (= the σ-order continuous dual of B(Σ, X)) is relatively σ(B(Σ, X)^*, B(Σ, X)^**)-sequentially compact. As a consequence, we obtain a Grothendieck type theorem saying that σ(B(Σ, X)^*, B(Σ, X))-convergent sequences in B(Σ, X)_c^~ are σ(B(Σ, X)^*, B(Σ, X)^**)-convergent.     

Additive maps between standard operator algebras compressing certain spectral functions

We characterize the surjective additive maps compressing the spectral function Δ（·） between standard operator algebras acting on complex Banach spaces, where Δ（·） stands for any one of nine spectral functions σ（·）, σl（·）, σr（·）,σl（·） ∩ σr（·）, δσ（·）, ησ（·）, σap（·）, σs（·）, and σap（·） ∩ σs（·）.     

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.     

Maps preserving the nilpotency of products of operators

Let B(X) be the algebra of all bounded linear operators on the Banach space X, and let N(X) be the set of nilpotent operators in B(X). Suppose ?:B(X)→B(X) is a surjective map such that A,B∈B(X) satisfy AB∈N(X) if and only if ?(A)?(B)∈N(X). If X is infinite dimensional, then there exists a map f:B(X)→C?{0} such that one of the following holds:

(a)

There is a bijective bounded linear or conjugate-linear operator S:X→X such that ? has the form A?S[f(A)A]S^-1.

(b)

The space X is reflexive, and there exists a bijective bounded linear or conjugate-linear operator S : X′ → X such that ? has the form A ? S[f(A)A′]S⁻¹.

If X has dimension n with 3 ? n < ∞, and B(X) is identified with the algebra M_n of n × n complex matrices, then there exist a map f:M_n→C?{0}, a field automorphism ξ:C→C, and an invertible S ∈ M_n such that ? has one of the following forms:

     

Maps preserving the idempotency of products of operators

Let B(X) be the algebra of all bounded linear operators on a complex Banach space X. We give the concrete form of every unital surjective map φ on B(X) such that AB is a non-zero idempotent if and only if φ(A)φ(B) is for all A,B∈B(X) when the dimension of X is at least 3.     

Ranges of positive contractive projections in Nakano spaces

We show that in any nontrivial Nakano space X=L_p(·) (Ω, Σ, μ) with essentially bounded random exponent function p(·), the range Y = R(P) of a positive contractive projection P is itself representable as a Nakano space L_pY(·) (Ω_Y Σ_Y, ν_Y), for a certain measurable set Y_Ω⊆Ω (the support of the range), a certain sub-sigma ring Y_Σ⊆Σ (with maximal element Ω_Y) naturally determined by the lattice structure of Y, and a semi-finite measure ν_Y, namely the restriction of some measure Ω on E which is equivalent to μ. Furthermore, we show that the random exponent p_Y(·) associated with such a range can be taken to be the restriction to Ω_Y of the random exponent p(·) (this restriction turns out to be Σ_Y-measurable). As an application of this result, we find Banach lattice isometric characterizations of suitable classes of Nakano spaces. These classes are defined in terms of an important lattice-isometric invariant of Nakano spaces, the essential range of the variable exponent.     

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.