Banach Journal of Mathematical Analysis最新文献

Let (mu ) be a Borel probability measure with compact support on ({mathbb {R}}), we say (mu ) is a spectral measure if there exists a countable set (Lambda subset {mathbb {R}}) such that the collection of exponential functions (E(Lambda ):={e^{-2pi ilangle lambda , xrangle }: lambda in Lambda }) forms an orthonormal basis for the Hilbert space (L^2(mu )). In this case, (Lambda ) is called a spectrum of (mu ). In this paper, we first characterize the spectral structure of self-similar spectral measures (mu _{t,D}) on ({mathbb {R}}), where D is a strict product-form digit set with respect to an integer b and t is an integer which has a proper factor b. And then we settle the spectral eigenvalue (or scaling spectrum) problem for the spectral measure (mu _{t,D}).

We introduce a class of Banach algebras that we call anti-C*-algebras. We show that the normed standard embedding of a C*-ternary ring is the direct sum of a C*-algebra and an anti-C*-algebra. We prove that C*-ternary rings and anti-C*-algebras are semisimple. We give two new characterizations of C*-ternary rings which are isomorphic to a TRO (ternary ring of operators), providing answers to a query raised by Zettl (Adv Math 48(2): 117–143, 1983), and we propose some problems for further study.

This paper is concerned with the convergence of power sequences and stability of Hilbert space operators, where “convergence” and “stability” are considered with respect to weak, strong and norm topologies. It is proved that an operator has a convergent power sequence if and only if it is a (not necessarily orthogonal) direct sum of an identity operator and a stable operator. This reduces the issue of convergence of the power sequence of an operator T to the study of stability of T. The question of when the limit of the power sequence is an orthogonal projection is investigated. Among operators sharing this property are hyponormal and contractive ones. In particular, a hyponormal or a contractive operator with no identity part is stable if and only if its power sequence is convergent. In turn, a unitary operator has a weakly convergent power sequence if and only if its singular-continuous part is weakly stable and its singular-discrete part is the identity. Characterizations of the convergence of power sequences and stability of subnormal operators are given in terms of semispectral measures.

It is known that every graph with n vertices embeds stochastically into trees with distortion (O(log n)). In this paper, we show that this upper bound is sharp for a large class of graphs. As this class of graphs contains Laakso graphs, this result extends known examples that obtain this largest possible stochastic distortion.

Let (P) be a property of (C^*)-algebras which may be satiesfied or not, and (mathscr {S}(P)) be the set of separable (C^*)-algebras which satiesfies (P). We give a sufficient condition for (P) to admit a universal separable element (Ain mathscr {S}(P)) in the sense that for any (Bin mathscr {S}(P)), there exists a surjective (*)-homomorphism (pi :Arightarrow B), and use the sufficient condition to show that when (P) is “unital with stable rank n”, “the small projection property” or “unital with stable exponential lenght b”, the sufficient condition is satisfied and hence there exists a corresponding universal (C^*)-algebra. We also give a stronger condition for property (P), which additionally implies that the set of corresponding universal (C^*)-algebras is uncountable, and use it to show that the set of universal unital separable (C^*)-algebras of stable rank n is uncountable as an example.

We introduce the notion of isometric envelope of a subspace in a Banach space, establishing its connections with several key elements: (a) we explore its relation to the mean ergodic projection on fixed points within a semigroup of contractions, (b) we draw parallels with Korovkin sets from the 1970s, (c) we investigate its impact on the extension properties of linear isometric embeddings. We use this concept to address the recent conjecture that the Gurarij space and the spaces (L_p), (p notin 2{mathbb {N}}+4) are the only separable approximately ultrahomogeneous Banach spaces (a certain multidimensional transitivity of the action of the linear isometry group). The similar conjecture for Fraïssé Banach spaces (a strengthening of the approximately homogeneous property) is also considered. We characterize the Hilbert space as the only separable reflexive space in which any closed subspace coincides with its envelope; and we show that the Gurarij space satisfies the same property. We compute some envelopes in the case of Lebesgue spaces, showing that the reflexive (L_p)-spaces are the only reflexive rearrangement invariant spaces on [0, 1] for which all 1-complemented subspaces are envelopes. We also identify the isometrically unique “full” quotient space of (L_p) by a Hilbertian subspace, for appropriate values of p, as well as the associated topological group embedding of the unitary group into the isometry group of (L_p).

Let (0<p<infty , alpha >-1,) and (beta ,gamma in {mathbb {R}}.) Let (mu ) be a finite positive Borel measure on the unit disk ({mathbb {D}}.) The Zygmund space (L^{p,beta }(dmu )) consists of all measurable functions f on ({mathbb {D}}) such that (|f|^plog ^beta (e+|f|)in L^1(dmu )) and the Bergman–Zygmund space (A^{p,beta }_{alpha }) is the set of all analytic functions in (L^{p,beta }(dA_alpha ),) where (dA_alpha =c_alpha (1-|z|^2)^alpha dA.) We prove an interpolation theorem for the Zygmund space assuming the weak type estimates on the Zygmund spaces themselves at the end points rather than the weak (L^p-L^q) type estimates at the end points. We show that the Bergman–Zygmund space is equal to the (log ^beta (e/(1-|z|)) dA_alpha (z)) weighted Bergman space as a set and characterize the bounded and compact Carleson measure (mu ) from (A^{p,beta }_{alpha }) into (A^{p,gamma }(dmu ),) respectively. The Carleson measure characterizations are of the same type for any pairs of ((beta , gamma )) whether (beta <gamma ) or (gamma le beta .)

We establish that the Volterra-type integral operator (J_b) on the Hardy spaces (H^p) of the unit ball ({mathbb {B}}^n) exhibits a rather strong rigid behavior. More precisely, we show that the compactness, strict singularity and (ell ^p)-singularity of (J_b) are equivalent on (H^p) for any (1 le p < infty ). Moreover, we show that the operator (J_b) acting on (H^p) cannot fix an isomorphic copy of (ell ^2) when (p ne 2.)

Let (X, d) be a compact metric space and ({mathcal {A}}) be a commutative and semisimple Banach algebra. Some of our recent works are related to the several (mathrm BSE) concepts of the vector-valued Lipschitz algebra (textrm{Lip}(X,{mathcal {A}})). In this paper as the main purpose, we verify the (mathrm BED) property for (textrm{Lip}(X,{mathcal {A}})), which is actually different from the (mathrm BSE) feature. We first prove as an elementary result that (textrm{Lip}(X,{mathcal {A}})) is regular if and only if ({mathcal {A}}) is so. Then we prove that ({mathcal {A}}) is a (mathrm BED) algebra, whenever (textrm{Lip}(X,{mathcal {A}})) is so. Afterwards, we verify the converse of this statement. Indeed, we prove that if ({mathcal {A}}) is a (mathrm BED) algebra then (C^{0}_{textrm{BSE}}(Delta (textrm{Lip}(X,{mathcal {A}})))subseteq widehat{textrm{Lip}(X,{mathcal {A}})}) and (widehat{textrm{Lip}Xotimes {mathcal {A}}}subseteq C^{0}_{textrm{BSE}}(Delta (textrm{Lip}(X,{mathcal {A}}))).) It follows that if (textrm{Lip}Xotimes {mathcal {A}}) is dense in (textrm{Lip}(X,{mathcal {A}})) then (textrm{Lip}(X,{mathcal {A}})) is a (mathrm BED) algebra, provided that ({mathcal {A}}) is so. Moreover, we conclude that the necessary and sufficient condition for the unital and in particular finite dimensional Banach algebra ({mathcal {A}}), to be a (mathrm BED) algebra is that (textrm{Lip}(X,{mathcal {A}})) is a (mathrm BED) algebra. Finally, regarding to some known results which disapproves the (mathrm BSE) property for (textrm{lip}_{alpha }(X,{mathcal {A}})) ((0<alpha <1)), we show that for any commutative and semisimple Banach algebra ({mathcal {A}}) with ({{mathcal {A}}}_0ne emptyset ), (textrm{lip}_{alpha }(X,{mathcal {A}})) fails to be a (mathrm BED) algebra, as well.

The main aim of the paper is to investigate some sufficient conditions which guarantee the convergence of sequences of positive linear operators towards composition operators within the framework of function spaces defined on a metric space. Among other things, the adopted approach allows to obtain a unifying reassessment of two milestones of the approximation theory by positive linear operators, namely, Korovkin’s theorem and Feller’s theorem together with some new extensions of them to the more general case where the limit operator is a composition operator. Some applications are shown and, among them, the convergence of Bernstein–Schnabl operator is enlightened in the framework of Banach spaces.