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.     

Unital and multiplicatively spectrum-preserving surjections between semi-simple commutative Banach algebras are linear and multiplicative

Let T be a surjective map from a unital semi-simple commutative Banach algebra A onto a unital commutative Banach algebra B. Suppose that T preserves the unit element and the spectrum σ(fg) of the product of any two elements f and g in A coincides with the spectrum σ(TfTg). Then B is semi-simple and T is an isomorphism. The condition that T is surjective is essential: An example of a non-linear and non-multiplicative unital map from a commutative C*-algebra into itself such that σ(TfTg)=σ(fg) holds for every f,g are given. We also show an example of a surjective unital map from a commutative C*-algebra onto itself which is neither linear nor multiplicative such that σ(TfTg)⊂σ(fg) holds for every f,g.     

Surjective linear maps preserving local spectra

Let B(X) be the algebra of all bounded linear operators on an infinite dimensional complex Banach space X. We characterize linear surjective and continuous maps on B(X) preserving different local spectral quantities at a nonzero fixed vector.     

Multiplicatively range-preserving maps between Banach function algebras

Let A and B be two Banach function algebras on locally compact Hausdorff spaces X and Y, respectively. Let T be a multiplicatively range-preserving map from A onto B in the sense that (TfTg)(Y)=(fg)(X) for all f,g∈A. We define equivalence relations on appropriate subsets and of X and Y, respectively, and show that T induces a homeomorphism between the quotient spaces of and by these equivalence relations. In particular, if all points in the Choquet boundaries of A and B are strong boundary points, then and are equal to the Choquet boundaries of A and B, respectively, and moreover, there exist a continuous function h on the Choquet boundary of B taking its values in {−1,1} and a homeomorphism φ from the Choquet boundary of B onto the Choquet boundary of A such that Tf(y)=h(y)f(φ(y)) for all f∈A and y in the Choquet boundary of B. For certain Banach function algebras A and B on compact Hausdorff spaces X and Y, respectively, we can weaken the surjectivity assumption and give a representation for maps belonging 2-locally to the family of all multiplicatively range-preserving maps from A onto B.     

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.     

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.     

Weighted composition operators between weighted spaces of vector-valued holomorphic functions on Banach spaces

Let B(E, F) be the Banach Space of all continuous linear operators from a Banach Space E into a Banach space F. Let U_X and U_Y be balanced open subsets of Banach spaces X and Y, respectively. Let V and W be two Nachbin families of continuous weights on U_X and U_Y, respectively. Let HV(U_X, E) (or HV₀(U_X, E)) and HW(U_Y, F) (or HW₀(U_Y, F)) be the weighted spaces of vector-valued holomorphic functions. In this paper, we investigate the holomorphic mappings ? : U_Y → U_X and ψ : U_Y → B(E, F) which generate weighted composition operators between these weighted spaces.     

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.     

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

     

Maps completely preserving idempotents and maps completely preserving square-zero operators

Let X, Y be real or complex Banach spaces with dimension greater than 2 and let A, B be standard operator algebras on X and Y, respectively. In this paper, we show that every map completely preserving idempotence from A onto B is either an isomorphism or (in the complex case) a conjugate isomorphism; every map completely preserving square-zero from A onto B is a scalar multiple of either an isomorphism or (in the complex case) a conjugate isomorphism.     

Lipschitz Quotients from Metric Trees and from Banach Spaces Containing ?1

If X is any separable Banach space containing l₁, then there is a Lipschitz quotient map from X onto any separable Banach space Y.     

Some well-posed functional equations which generate semigroups

Consider the abstract linear functional equation (FE) (Dx)(t) = f(t) (t ? 0), x(t) = ?(t) (t ? 0) in a Banach space B. A theorem is proven which contains the following result as a special case. Let Y(R; B; η) be a L^p-space or C₀-space on R = (?t8, ∞), with a suitable weight function η, and with values in B. Let D be a closed (unbounded) causal linear operator in Y(R; B; η), which commutes with translations. Suppose that D + λI has a continuous causal inverse for some complex λ, and that D restricted to those functions in Y(R;B;η) which vanish on R^? = (?∞, 0] has a continuous causal inverse. Then (FE) generates a strongly continuous semigroup of translation type on a Banach space, which is essentially the cross product of the restriction of the domain of D to R^? and Y(R⁺; B; η). Examples with B = Cⁿ on how the theory applies to a neutral functional differential equation, a difference equation, a Volterra integrodifferential equation (with nonintegrable kernel but integrable resolvent), and a fractional order functional differential equation are given. Also, an abstract neutral functional differential equation in a Hilbert space is studied and applications to an abstract Volterra integrodifferential equation in a Banach space are indicated.     

Sufficient enlargements of minimal volume for finite-dimensional normed linear spaces

Let BY denote the unit ball of a normed linear space Y. A symmetric, bounded, closed, convex set A in a finite-dimensional normed linear space X is called a sufficient enlargement for X if, for an arbitrary isometric embedding of X into a Banach space Y, there exists a linear projection such that P(BY)⊂A. The main results of the paper: (1) Each minimal-volume sufficient enlargement is linearly equivalent to a zonotope spanned by multiples of columns of a totally unimodular matrix. (2) If a finite-dimensional normed linear space has a minimal-volume sufficient enlargement which is not a parallelepiped, then it contains a two-dimensional subspace whose unit ball is linearly equivalent to a regular hexagon.     

On the product of relatively weakly almost Lindelöf subsets

The purpose of this note is to show that there exist two Tychonoff spaces X, Y, a subset A of X and a subset B of Y such that A is weakly almost Lindelöf in X and B is weakly almost Lindelöf in Y, but A × B is not weakly almost Lindelöf in X × Y.     

Banach spaces in various positions

We formulate a general theory of positions for subspaces of a Banach space: we define equivalent and isomorphic positions, study the automorphy index a(Y,X) that measures how many non-equivalent positions Y admits in X, and obtain estimates of a(Y,X) for X a classical Banach space such as ?p,Lp,L₁,C(ωω) or C[0,1]. Then, we study different aspects of the automorphic space problem posed by Lindenstrauss and Rosenthal; namely, does there exist a separable automorphic space different from c₀ or ?₂? Recall that a Banach space X is said to be automorphic if every subspace Y admits only one position in X; i.e., a(Y,X)=1 for every subspace Y of X. We study the notion of extensible space and uniformly finitely extensible space (UFO), which are relevant since every automorphic space is extensible and every extensible space is UFO. We obtain a dichotomy theorem: Every UFO must be either an L_∞-space or a weak type 2 near-Hilbert space with the Maurey projection property. We show that a Banach space all of whose subspaces are UFO (called hereditarily UFO spaces) must be asymptotically Hilbertian; while a Banach space for which both X and X^∗ are UFO must be weak Hilbert. We then refine the dichotomy theorem for Banach spaces with some additional structure. In particular, we show that an UFO with unconditional basis must be either c₀ or a superreflexive weak type 2 space; that a hereditarily UFO Köthe function space must be Hilbert; and that a rearrangement invariant space UFO must be either L_∞ or a superreflexive type 2 Banach lattice.     

-Operator frames for a Banach space

In this paper, (p,Y)-Bessel operator sequences, operator frames and (p,Y)-Riesz bases for a Banach space X are introduced and discussed as generalizations of the usual concepts for a Hilbert space and of the g-frames. It is proved that the set of all (p,Y)-Bessel operator sequences for a Banach space X is a Banach space and isometrically isomorphic to the operator space B(X,ℓ^p(Y)). Some necessary and sufficient conditions for a sequence of operators to be a (p,Y)-Bessel operator sequence are given. Also, a characterization of an independent (p,Y)-operator frame for X is obtained. Lastly, it is shown that an independent (p,Y)-operator frame for X is just a (p,Y)-Riesz basis for X and has a unique dual (q,Y^*)-operator frame for X^*.     

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.     

Operators on the space of vector-valued totally measurable functions

Let L(X,Y) stand for the space of all bounded linear operators between real Banach spaces X and Y, and let Σ be a σ-algebra of sets. A bounded linear operator T from the Banach space B(Σ,X) of X-valued Σ-totally measurable functions to Y is said to be σ-smooth if ‖T(fn)Y_‖→0 whenever a sequence of scalar functions (‖fn(⋅)X_‖) is order convergent to 0 in B(Σ). It is shown that a bounded linear operator is σ-smooth if and only if its representing measure is variationally semi-regular, i.e., as An↓∅ (here stands for the semivariation of m on A∈Σ). As an application, we show that the space Lσs(B(Σ,X),Y) of all σ-smooth operators from B(Σ,X) to Y provided with the strong operator topology is sequentially complete. We derive a Banach-Steinhaus type theorem for σ-smooth operators from B(Σ,X) to Y. Moreover, we characterize countable additivity of measures in terms of continuity of the corresponding operators .     

Operators whose dual has non-separable range

Let X and Y be separable Banach spaces and T:X→Y be a bounded linear operator. We characterize the non-separability of T^?(Y^?) by means of fixing properties of the operator T.