Advances in Operator Theory最新文献

We prove a Lie–Trotter type formula for q-exponential operators:

where E is a Banach space, U, V bounded linear operators on E and (qin mathbb {R}), (qne 1).

New characterizations of left and right generalized Drazin inverses are presented in Banach algebra using tripotents. We explore Cline’s formula for those inverses. Also, we find the conditions under which the product of two left (or right) generalized Drazin invertible elements has the appropriate one-sided generalized Drazin inverse. Consequently, we study when the sum of two zero product elements has a one-sided generalized Drazin inverse. Perturbation results for the left (or right) generalized Drazin inverse are also investigated. Applying our results, we obtain new properties of the left (or right) Drazin inverse and recover known results about the generalized Drazin inverse.

In this work, we introduce and study generalized entropy numbers for sets and operators acting on Banach spaces. The classical notion of Hausdorff entropy numbers becomes a particular case of the given definition. We also provide several other examples of generalized entropy numbers for sets and operators. We prove several properties for the general case.

In this review paper we study non-compact operators and embeddings between function spaces, highlighting interesting phenomena and the significance of Bernstein numbers. In particular, we demonstrate that for non-compact maps the usual s-numbers (e.g., approximation, Kolmogorov, and entropy numbers) fail to reveal finer structural properties, and one must instead consider concepts such as strict singularity and Bernstein numbers.

We study the action of the nonlinear mapping G[z] between real or complex Banach spaces in the vicinity of a given curve with respect to possible linearization, emerging patterns of level sets, as well as existing solutions of (G[z]=0). The results represent local generalizations of the standard implicit or inverse function theorem and of Newton’s Lemma, considering the order of approximation needed to obtain solutions of (G[z]=0). The main technical tool is given by Jordan chains with increasing rank, used to obtain an Ansatz, appropriate for transformation of the nonlinear system to its linear part. The family of linear mappings is restricted to the case of an isolated singularity. Geometrically, the Jordan chains define a generalized cone around the given curve, composed of approximate solutions of order 2k with k denoting the maximal rank of Jordan chains needed to ensure k-surjectivity of the linear family. Along these lines, the zero set of G[z] in the cone is calculated immediately, agreeing up to the order of (k-1) with the given approximation. Hence, the results may also be interpreted as a version of Tougeron’s implicit function theorem in Banach spaces, essentially restricted to the arc case of a single variable. Finally, by considering a left shift of the Jordan chains, the Ansatz can be modified in a systematic way to obtain a sequence of refined versions of linearization theorems and Newton Lemmas in Banach spaces.

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.