Advances in Operator Theory最新文献

Let ({mathcal {X}}) be a complex Banach space, and let ({mathcal {B}}({mathcal {X}})) be the algebra of all bounded linear operators on ({mathcal {X}}.) In this paper, we characterize the general forms of surjective maps on ({mathcal {B}}({mathcal {X}})) that preserve the dimension of fixed points of Jordan triple product of operators.

Given a metrizable space Z,  denote by (operatorname {PM}(Z)) the space of continuous bounded pseudometrics on Z,  and denote by (operatorname {AM}(Z)) the one of continuous bounded admissible metrics on Z,  both of which are equipped with the sup-norm (Vert cdot Vert .) In this paper, we shall prove Banach–Stone type theorems on spaces of metrics, that is, for metrizable spaces X and Y,  X and Y are homeomorphic if and only if there exists a surjective isometry (T: operatorname {PM}(X) rightarrow operatorname {PM}(Y)) ((T: operatorname {AM}(X) rightarrow operatorname {AM}(Y))) satisfying some conditions. Then for each surjective isometry T,  there is a homeomorphism (phi : Y rightarrow X) such that for any (d in operatorname {PM}(X)) and for any (x, y in Y,) (T(d)(x,y) = d(phi (x),phi (y)).) Except for the case where the cardinality of X or Y is equal to 2,  the homeomorphism (phi ) can be chosen uniquely.

In this paper, we investigate operator-valued frames (OPV-frames) for phase (norm) retrieval. Firstly, we give a sufficient and necessary condition for phase retrievable OPV-frames in real finite-dimensional Hilbert spaces. Some conditions which are equivalent to phase retrievable OPV-frames are also presented. Secondly, we obtain some equivalent conditions to the norm retrievable OPV-frame in real and complex finite-dimensional Hilbert spaces. Finally, we show that the property of phase retrievable for real Hilbert spaces is stable under small perturbation of an OPV-frame. It is also shown that the property of norm retrievability is stable under enough small perturbations of an OPV-frame only for phase retrievable OPV-frames.

This paper is devoted to studying the behaviors of strongly singular Calderón–Zygmund operators T and their commutators [b, T] generated by T with (bin L_{loc}({mathbb {R}}^n)) on the Musielak–Orlicz Hardy spaces. The authors obtain the boundedness of T from the Musielak–Orlicz Hardy spaces (H^varphi ({mathbb {R}}^n)) to the Musielak–Orlicz spaces (L^varphi ({mathbb {R}}^n),) and from the Musielak–Orlicz Hardy spaces (H^varphi ({mathbb {R}}^n)) to themselves if (T^*1=0.) Meanwhile, the corresponding mapping properties for the commutators [b, T] are also obtained, provided that b belongs to (mathcal {BMO}_{varphi ,u}({mathbb {R}}^n),) a non-trivial subspace of ({textrm{BMO}}({mathbb {R}}^n).)

We extend the spectral theory of commutative C*-categories to the non-full case, introducing a suitable notion of spectral spaceoid providing a duality between a category of “non-trivial” (*)-functors of non-full commutative C*-categories and a category of Takahashi morphisms of “non-full spaceoids” (here defined). As a byproduct we obtain a spectral theorem for a non-full generalization of imprimitivity Hilbert C*-bimodules over commutative unital C*-algebras via continuous sections vanishing at infinity of a Hilbert C*-line-bundle over the graph of a homeomorphism between open subsets of the corresponding Gel’fand spectra of the C*-algebras.

As a main research area in applied and computational harmonic analysis, the theory and applications of framelets have been extensively investigated. Most existing literature is devoted to framelet systems that only use one dilation matrix as the sampling factor. To keep some key properties such as directionality, a framelet system often has a high redundancy rate. To reduce redundancy, a one-dimensional tight framelet with mixed dilation factors has been introduced for image processing. Though such tight framelets offer good performance in practice, their theoretical properties are far from being well understood. In this paper, we will systematically investigate framelets with mixed dilation factors, with arbitrary multiplicity in arbitrary dimensions. We will first study the discrete framelet transform employing a filter bank with mixed dilation factors and discuss its various properties. Next, we will introduce the notion of a discrete affine system in (l_{2}(mathbb {Z}^d)) and study discrete framelet transforms with mixed dilation factors. Finally, we will discuss framelets and wavelets with mixed dilation factors in the space (L_{2}(mathbb {R}^d)).

The differences of integration-composition operators on various analytic function spaces have attracted lots of attention for decades. In this note, we study the differences of mixed products of Volterra operators and composition operators on Bloch spaces. To be specific, we characterize the following four types of differences of mixed products: (I_gC_{varphi }-C_{psi }J_h), (J_gC_{varphi }-C_{psi }I_h), (I_gC_{varphi }-C_{psi }I_h) and (J_gC_{varphi }-C_{psi }J_h). One surprising result is that unbounded (I_gC_{varphi }) and (C_{psi }J_h) can not induce bounded difference (I_gC_{varphi }-C_{psi }J_h).

This paper explores the properties of two novel operators, the 2-Laurent operator and the 2-Toeplitz operator, acting on the Lebesgue space and the Hardy space, respectively. These operators are characterized by matrices with alternating constant entries along each diagonal parallel to the main diagonal. They are significant because they reduce to the classical Laurent and Toeplitz operators as special cases. The paper provides several important results, including characterizations of these operators, and also introduces generalizations for any natural number (k ge 2). These generalizations, termed the k-Laurent and k-Toeplitz operators, encompass the 2-Laurent and 2-Toeplitz operators as particular instances when (k=2).

We solve Gleason’s problem for harmonic mixed norm spaces (B^{p,q}_alpha (Omega ),) where (p,q ge 1, alpha >0,) on bounded star-shaped domains in ({mathbb {R}}^n,) in a slightly more general form than the standard one, specifically for higher-order derivatives. The approach here is functional analytic, in particular, we first discuss the boundedness of certain integral operators, and then we prove that Gleason’s problem is solvable on harmonic mixed norm spaces (B^{p,q}_alpha (Omega ).)

In this paper, we introduce a new family of function spaces of Besov-Triebel-Lizorkin type. We present the (varphi )-transform characterization of these spaces in the sense of Frazier and Jawerth and we prove their Sobolev and Franke-Jewarth embeddings. Also, we establish the smooth atomic, molecular and wavelet decomposition of these function spaces. Characterizations by ball means of differences are given. Finally, we investigate a series of examples which play an important role in the study of function spaces of Besov-Triebel-Lizorkin type.