Annals of Functional Analysis最新文献

For tuples of compact operators (mathcal {T}=(T_1,ldots , T_d)) and (mathcal {S}=(S_1,ldots ,S_d)) on Banach spaces over a field (mathbb {F}), considering the joint p-operator norms on the tuples, we study (dist(mathcal {T},mathbb {F}^dmathcal {S}),) the distance of (mathcal {T}) from the d-dimensional subspace (mathcal {F}^dmathcal {S}:={{textbf {z}}mathcal {S}:{textbf {z}}in mathbb {F}^d}.) We obtain a relation between (dist(mathcal {T},mathbb {F}^dmathcal {S})) and (dist(T_i,mathbb {F}S_i),) for (1le ile d.) We prove that if (p=infty ,) then (dist(mathcal {T},mathbb {F}^dmathcal {S})=underset{1le ile d}{max }dist(T_i,mathbb {F}S_i),) and for (1le p<infty ,) under a sufficient condition, (dist(mathcal {T},mathbb {F}^dmathcal {S})^p=underset{1le ile d}{sum }dist(T_i,mathbb {F}S_i)^p.) As a consequence, we deduce the equivalence of Birkhoff-James orthogonality, (mathcal {T}perp _B mathbb {F}^dmathcal {S} Leftrightarrow T_iperp _B S_i,) under a sufficient condition. Furthermore, we explore the relation of one sided Gâteaux derivatives of (mathcal {T}) in the direction of (mathcal {S}) with that of (T_i) in the direction of (S_i.) Applying this, we explore the relation between the smoothness of (mathcal {T}) and (T_i.) By identifying an operator, whose range is (ell _infty ^d,) as a tuple of functionals, we effectively use the results obtained here for operators whose range is (ell _infty ^d) and deduce nice results involving functionals.

We study non-unital (C^*)-algebras such that for any element, there exists a local unit and prove that in such algebras there are no frames. This fact was previously known only for commutative algebras. Among other results, we establish some necessary properties of frames in (C^*)-algebras (which are of independent interest in the noncommutative topology), and consider several examples of (C^*)-algebras that are new in this context.

In this paper, we establish the weak factorizations of the Hardy space associated with the Dunkl operator via the bilinear forms of Dunkl–Riesz transforms ({{mathcal {R}}_{j}}_{j=1}^{d}.) Note that the kernels of ({{mathcal {R}}_{j}}_{j=1}^{d}) involve both the Euclidean and the Dunkl metrics, which are not equivalent. As an application, we provide a new proof for the sufficiency of characterization of the ({textrm{BMO}}) space associated to the Dunkl operator via the commutators of ({{mathcal {R}}_{j}}_{j=1}^{d}.)

This paper studies the invariant subspaces of the Bergman shift using the resolvent set analysis approach introduced by Douglas and Yang. We construct invariants on the resolvent set of the Bergman shift to describe the Bergman inner functions and the inclusion relationship of invariant subspaces of the Bergman shift. We generalize the concept of power sets, originally introduced by Douglas and Yang for quasinilpotent operators, to any boundary point of the spectrum of any operator. We compute the newly defined power sets for the conjugate of a class of compression operators of the Bergman shift, and demonstrate how they reflect the structure of invariant subspaces.

In the present paper, the notion of weak degree of nondensifiability, w-DND, is introduced. Likewise, we analyze its main properties, and we also prove that the w-DND is actually an upper bound for any measure of weak noncompactness. Moreover, for the De Blasi measure of weak noncompactness, such an upper bound is sharp. As an application of our results, we characterize both Schur and Dunford–Pettis properties of a Banach space in terms of the w-DND, which turns out this new concept into a useful tool in functional analysis.

Let X be a Banach space such that there exists a Banach space (^*X) satisfying (( ^*X )^ *= X). In this paper, we introduce X-valued Bourgain–Morrey spaces. We show that (^*X)-valued block spaces are the predual of X-valued Bourgain–Morrey spaces. We obtain the completeness, denseness, and Fatou property of (^*X)-valued block spaces. We give a description of the dual of X-valued Bourgain–Morrey spaces and conclude the reflexivity of these spaces. The boundedness of powered Hardy–Littlewood maximal operator in vector-valued block spaces is obtained.

We study when an additive mapping preserving orthogonality between two complex inner product spaces is automatically complex-linear or conjugate-linear. Concretely, let H and K be complex inner product spaces with (hbox{dim}(H)ge 2), and let (A: Hrightarrow K) be an additive map preserving orthogonality. We obtain that A is zero or a positive scalar multiple of a real-linear isometry from H into K. We further prove that the following statements are equivalent:

(a):

A is complex-linear or conjugate-linear.

(b):

For every (zin H) we have (A(i z) in {pm i A(z)}).

(c):

There exists a non-zero point (zin H) such that (A(i z) in {pm i A(z)}).

(d):

There exists a non-zero point (zin H) such that (i A(z) in A(H)).

The mapping A is neither complex-linear nor conjugate-linear if, and only if, there exists a non-zero (xin H) such that (i A(x)notin A(H)) (equivalently, for every non-zero (xin H), (i A(x)notin A(H))). Among the consequences, we show that, under the hypothesis above, the mapping A is automatically complex-linear or conjugate-linear if A has dense range, or if H and K are finite dimensional with (hbox{dim}(K)< 2hbox{dim}(H)).

We identify the smooth points of (L^1(mu ,X)), and provide some necessary and sufficient conditions for left and right symmetry of points with respect to Birkhoff–James orthogonality in (L^p(mu ,X), 1le p<infty), where (mu) is any complete positive measure and X is a Banach space with some suitable properties.

In this paper, we extend Ando’s theorem on paranormal operators, which states that if (T in mathfrak {B}(mathcal {H})) is a paranormal operator and there exists (n in mathbb {N}) such that (T^n) is normal, then (T) is normal. We generalize this result to the broader classes of (k)-paranormal operators and absolute-(k)-paranormal operators. Furthermore, in the case of a separable Hilbert space (mathcal {H}), we show that if (T in mathfrak {B}(mathcal {H})) is a (k)-quasi-paranormal operator for some (k in mathbb {N}), and there exists (n in mathbb {N}) such that (T^n) is normal, then (T) decomposes as (T = T' oplus T''), where (T') is normal and (T'') is nilpotent of nil-index at most (min {n,k+1}), with either summand potentially absent.

In the present paper, we study a class of integral operators of probabilistic type, which are constructed by means of the generalized Gamma distribution. In particular, we discuss their approximation properties in weighted continuous function spaces and (L^p)-spaces, providing some estimates of the rate of convergence by means of different moduli of smoothness as well as an asymptotic formula. The paper concludes with some illustrative examples.