Open Problems In Theory Research Articles

Geraghty–Pata–Suzuki-Type Proximal Contractions and Related Coincidence Best Proximity Point Results

The objective of this research paper is to establish the existence and uniqueness of the best proximity and coincidence with best proximity point results, specifically focusing on Geraghty–Pata–Suzuki-type proximal mappings. To achieve this, we introduce three types of mappings, all within the context of a complete metric space: an α-θ-Geraghty–Pata–Suzuki-type proximal contraction; an α-θ-generalized Geraghty–Pata–Suzuki-type proximal contraction; and an α-θ-modified Geraghty–Pata–Suzuki-type proximal contraction. These new results generalize, extend, and unify various results from the existing literature. Symmetry plays a crucial role in solving nonlinear problems in operator theory, and the variables involved in the metric space are symmetric. Several illustrative examples are provided to showcase the superiority of our results over existing approaches.

Sharpening Bounds for Multilinear Schur Multipliers

In this paper we make a brief survey of major estimates for norms and traces of multilinear Schur multipliers. We also discuss several open problems in operator theory and mathematical physics whose resolution could be reduced to sharpening existing bounds for Schur multipliers.

Berezin number inequalities via convex functions

The Berezin symbol ?A of an operator A on the reproducing kernel Hilbert space H (?) over some set ? with the reproducing kernel k? is defined by ? (?) = ?A k?/||k?||, k?/||k?||?, ? ? ?. The Berezin number of an operator A is defined by ber(A) := sup ??? |?(?)|. We study some problems of operator theory by using this bounded function ?, including treatments of inner product inequalities via convex functions for the Berezin numbers of some operators. We also establish some inequalities involving of the Berezin inequalities.

Positive answer to the invariant and hyperinvariant subspaces problems for hyponormal operators

Abstract The question whether every operator on infinite-dimensional Hilbert space 𝐻 has a nontrivial invariant subspace or a nontrivial hyperinvariant subspace is one of the most difficult problems in operator theory. This problem is open for more than half a century. A subnormal operator has a nontrivial invariant subspace, but the existence of nontrivial invariant subspace for a hyponormal operator 𝑇 still open. In this paper we give an affirmative answer of the existence of a nontrivial hyperinvariant subspace for a hyponormal operator. More generally, we show that a large classes of operators containing the class of hyponormal operators have nontrivial hyperinvariant subspaces. Finally, every generalized scalar operator on a Banach space 𝑋 has a nontrivial invariant subspace.

Scala step-by-step: soundness for DOT with step-indexed logical relations in Iris

The metatheory of Scala’s core type system—theDependent Object Types (DOT)calculus—is hard to extend, like the metatheory of other type systems combining subtyping and dependent types. Soundness of important Scala features therefore remains an open problem in theory and in practice. To address some of these problems, we use asemantics-firstapproach to develop a logical relations model for a new version of DOT, calledguarded DOT (gDOT). Our logical relations model makes use of an abstract form ofstep-indexing, as supported by the Iris framework, to model various forms of recursion in gDOT. To demonstrate the expressiveness of gDOT, we show that it handles Scala examples that could not be handled by previous versions of DOT, and prove using our logical relations model that gDOT provides the desired data abstraction. The gDOT type system, its semantic model, its soundness proofs, and all examples in the paper have been mechanized in Coq.

Joint numerical ranges: recent advances and applications minicourse by V. Müller and Yu. Tomilov

AbstractWe present a survey of some recent results concerning joint numerical ranges ofn-tuples of Hilbert space operators, accompanied with several new observations and remarks. Thereafter, numerical ranges techniques will be applied to various problems of operator theory. In particular, we discuss problems concerning orbits of operators, diagonals of operators and their tuples, and pinching problems. Lastly, motivated by known results on the numerical radius of a single operator, we examine whether, given bounded linear operatorsT1, . . .,Tnon a Hilbert spaceH, there exists a unit vectorx ∈ Hsuch that|〈Tjx,x〉|is “large” for allj= 1, . . . ,n.

Perturbation bounds for eigenspaces under a relative gap condition

A basic problem in operator theory is to estimate how a small perturbation affects the eigenspaces of a self-adjoint compact operator. In this paper, we prove upper bounds for the subspace distance, tailored for relative perturbations. As a main example, we consider the empirical covariance operator and show that a sharp bound can be achieved under a relative gap condition. The proof is based on a novel contraction phenomenon, contrasting previous spectral perturbation approaches.

On some problems for operators on the reproducing kernel Hilbert space

ABSTRACT In this study, some problems of operator theory on the reproducing kernel Hilbert space by using the Berezin symbols method are investigated. Namely, invariant subspaces of weighted composition operators on are studied. Moreover, some new inequalities for the Berezin number of operators are proved. In particular, new reverse inequalities for the Berezin numbers and of operators and A on the reproducing kernel Hilbert space are given. Also, reverse inequalities for the Berezin number of two operators are proved. Under some conditions we prove the power inequality which is related to a well known analogue estimate of Halmos for the numerical radius.

Halmos problems and related results in the theory of invariant subspaces

This paper contains a survey of main results in the theory of invariant subspaces that have a direct or intermediate connection with the well-known list of problems in operator theory posed by Halmos in 1970. For most results we present brief proofs, schemes of proofs, or basic ideas of proofs. Bibliography: 124 titles.

Progress in the open problems in theory of functions of bounded index

Progress in the open problems in theory of functions of bounded index

A new characterization of subnormality for a class of 2-variable weighted shifts with 1-atomic core

Given a pair T≡(T1,T2) of commuting subnormal Hilbert space operators, the Lifting Problem for Commuting Subnormals (LPCS) calls for necessary and sufficient conditions for the existence of a commuting pair N≡(N1,N2) of normal extensions of T1 and T2. This is an old problem in operator theory. The aim of this paper is to study LPCS. There are three well-known subnormal characterizations for operators: the Berger Theorem, the Bram–Halmos characterization, and Franks' result. In our paper, we study a new subnormal characterization which is related to these three well-known ones for a class of 2-variable weighted shifts. Thus, we can provide a large nontrivial class of 2-variable weighted shifts in which k-hyponormal (some k≥1) and subnormal are equal and the class is invariant under the action (h,ℓ)↦T(h,ℓ):=(T1h,T2ℓ) (h,ℓ≥1).

The Square Root Problem and Aluthge transforms of weighted shifts

AbstractIn this paper we consider the following question. When does there exist a square root of a probability measure supported on ? This question is naturally related to subnormality of weighted shifts. The main result of this paper is that if μ is a finitely atomic probability measure having at most 4 atoms, then μ has a square root, i.e., there exists a measure ν such that (* means the convolution) if and only if the Aluthge transform of a subnormal weighted shift with Berger measure μ is subnormal. As an application of them, we give non‐trivial, large classes of probability measures having a square root. We also prove that there are 6 and 7‐atomic probability measures which don't have any square root. Our results have a connection to the following long‐open problem in Operator Theory: characterize the subnormal operators having a square root.

Invariance of the order and type of a sequence of operators

In the paper, the invariance property of characteristics (the order and type) of an operator and of a sequence of operators with respect to a topological isomorphism is proved. These characteristics give precise upper and lower bounds for the expressions ‖An(x)‖p and enable one to state and solve problems of operator theory in locally convex spaces in a general setting. Examples of such problems are given by the completeness problem for the set of values of a vector function in a locally convex space, the structure problem for a subspace invariant with respect to an operator A, the problem of applicability of an operator series to a locally convex space, the theory of holomorphic operator-valued functions, the theory of operator and differential-operator equations in nonnormed spaces, and so on. However, the immediate evaluation of characteristics of operators (and of sequences of operators) directly by definition is practically unrealizable in spaces with more complicated structure than that of countably normed spaces, due to the absence of an explicit form of seminorms or to their complicated structure. The approach that we use enables us to find characteristics of operators and sequences of operators using the passage to the dual space, by-passing the definition, and makes it possible to obtain bounds for the expressions ‖An(x)‖p even if an explicit form of seminorms is unknown.

Diagonality and Idempotents with applications to problems in operator theory and frame theory

We prove that a nonzero idempotent is zero-diagonal if and only if it is not a Hilbert-Schmidt perturbation of a projection, along with other useful equivalences. Zero-diagonal operators are those whose diagonal entries are identically zero in some basis. We also prove that any bounded sequence appears as the diagonal of some idempotent operator, thereby providing a characterization of inner products of dual frame pairs in infinite dimensions. Furthermore, we show that any absolutely summable sequence whose sum is a positive integer appears as the diagonal of a finite rank idempotent.

Strongly Irreducible Operators and Indecomposable Representations of Quivers on Infinite-Dimensional Hilbert Spaces

We study several classes of indecomposable representations of quivers on infinite-dimensional Hilbert spaces and their relation. Many examples are constructed using strongly irreducible operators. Some problems in operator theory are rephrased in terms of representations of quivers. We shall show two kinds of constructions of quite non-trivial indecomposable Hilbert representations (H, f) of the Kronecker quiver such that \({End(H,f) = \mathbb{C} I}\) which is called transitive. One is a perturbation of a weighted shift operator by a rank-one operator. The other one is a modification of an unbounded operator used by Harrison,Radjavi and Rosenthal to provide a transitive lattice.

Reflexive operator algebras on Banach spaces

In this paper we study the reflexivity of a unital strongly closed algebra of operators with complemented invariant subspace lattice on a Banach space. We prove that if such an algebra contains a complete Boolean algebra of projections of finite uniform multiplicity and with the direct sum property, then it is reflexive, i.e., it contains every operator that leaves invariant every closed subspace in the invariant subspace lattice of the algebra. In particular, such algebras coincide with their bicommutant. Let A B.X/ denote a strongly closed algebra of operators on the Banach space X. Suppose that A has the property that each of its invariant subspaces has an invariant complement. If A contains a complete Boolean algebra of projections of finite uniform multiplicity and with the direct sum property as defined below, we prove that A is reflexive in the sense that it contains all the operators which leave its closed invariant subspaces invariant (Theorem 15). In particular such an algebra is equal to its bicommutant A 00 (Corollary 22). The problem of whether a strongly closed algebra of operators with complemented invariant subspace lattice is reflexive started to be studied in the sixties. This problem is a generalization of the invariant subspace problem in operator theory. Arveson [1967] introduced a technique for studying the particular case of transitive algebras on Hilbert spaces, namely the strongly closed algebras of operators on Hilbert spaces that have no nontrivial closed invariant subspaces. He proved that every transitive algebra that contains a maximal abelian von Neumann algebra coincides with the full algebra B.X/ if X is a complex Hilbert space. Douglas and Pearcy [1972] extended the result of Arveson to the case of transitive operator algebras containing an abelian von Neumann algebra of finite multiplicity. Hoover [1973] extended the result of Douglas and Pearcy to the case of reductive operator algebras on Hilbert spaces that contain C. Peligrad and M. Peligrad were supported in part by a Charles Phelps Taft Memorial Fund grant. M. Peligrad was also supported by NSF grant DMS-1208237. MSC2010: primary 47A15, 47B48; secondary 47C05.

The Paulsen Problem in operator theory

The Paulsen Problem in Hilbert space frame theory has proved to be one of the most intractable problems in the field. We will help explain why by showing that this problem is equivalent to a fundamental, deep problem in operator theory. This answers a question posed by Bodmann and Casazza. We will also give generalizations of these problems and we will spell out exactly the complementary versions of the problem. Mathematics subject classification (2010): 42C15, 46C05.

Generalizations of Bohr inequality for Hilbert space operators

Let B ( H ) be the space of all bounded linear operators on a complex separable Hilbert space H . Bohr inequality for Hilbert space operators asserts that for A , B ∈ B ( H ) and p , q > 1 real numbers such that 1 / p + 1 / q = 1 , | A + B | 2 ⩽ p | A | 2 + q | B | 2 with equality if and only if B = ( p − 1 ) A . In this paper, a number of generalizations of Bohr inequality for operators in B ( H ) are established. Moreover, Bohr inequalities are extended to multiple operators and some related inequalities are obtained. The results in this paper generalize results known so far. The idea of transforming problems in operator theory to problems in matrix theory, which are easy to handle, is the key role.

Parameter depending almost monotonic functions and their applications to dimensions in metric measure spaces

In connection with application to various problems of operator theory, we study almost monotonic functions w(x, r) depending on a parameter x which runs a metric measure space X, and the so called index numbers m(w, x), M(w, x) of such functions, and consider some generalized Zygmund, Bary, Lozinskii and Stechkin conditions. The main results contain necessary and sufficient conditions, in terms of lower and upper bounds of indices m(w, x) and M(w, x) , for the uniform belongness of functions w(·, r) to Zygmund‐Bary‐Stechkin classes. We give also applications to local dimensions in metric measure spaces and characterization of some integral inequalities involving radial weights and measures of balls in such spaces.

Preface

The Fifth Congress of the International Society for Theoretical Chemical Physics was held at the Renaissance Arts Hotel in New Orleans, Louisiana, during the period July 20–26, 2005. The timing was fortunate; less than 5 weeks after the meeting ended, the city of New Orleans was devastated by Hurricane Katrina and its aftermath. During recent years, great progress has been made in both theory and experiment in chemical phenomena. Significant developments have taken place in the area of large-scale correlation calculations, as well as in density functional theory. Novel scientific applications to nanomaterials, molecular biophysics, surfaces, polymers, etc., using both quantum and statistical mechanics methods are on a dramatic rise. Today the theory of chemistry (theoretical chemical physics) has many different branches, and it is understandable that communication between scientists in the field should and must improve. The main goal of the International Society for Theoretical Chemical Physics, established by Janos Ladik in 1990, is to provide a forum for researchers working in these different disciplines, by organizing an international congress every third year. The strongly interdisciplinary nature of the Society should stimulate progress and strongly promote the development of the whole field of theoretical chemical physics. Promote interaction, exchange of ideas and collaboration on an international level. Provide opportunities for students and postdoctoral research associates to interact with leaders in their fields and to present and discuss their own work. There were nine scientific sessions, each focusing on a different topic and organized by an internationally recognized leader in the field, who started the session with a “perspectives/overview” presentation. In addition, there were three poster sessions, involving all areas of chemical physics, materials science, and molecular biophysics. The nine sessions and the speakers were: Methodology: R. J. Bartlett (organizer), L. Adamowicz, W. D. Allen, S. Hirata, P. E. Hoggan, K. Jankowski, M. Nooijen, P. Piecuch, G. E. Scuseria. Statistical Mechanics: H. J. Kreuzer (organizer), S. Bonev, R. R. Netz, T. Seideman. Noncovalent Interactions: P. Hobza (organizer), M. Havenith, B. Jeziorski, K. Jordan, K. Müller-Dethlefs, M. S. de Vries, T. Zwier. Nanomaterials: J. M. Seminario (organizer), P. B. Balbuena, M. J. Frisch, G. S. Hwang, D. R. Salahub, D. Woolard. Surfaces and Materials: A. M. Chaka (organizer), L. Gagliardi, J. D. Gale, W. A. Goddard III, S. Grabowski, A. E. Masunov, S. L. Price, P. Vashishta, Y. A. Wang. Molecular Biophysics: H. Weinstein (organizer), J. Gao, D. Millar, N. Richards, W. A. Sokalski. Solvation: M. Orozco (organizer), C. L. Brooks III, C. J. Cramer, A. Klamt, B. Mennucci. Density Functional Theory: M. Levy (organizer), P. Ayers, K. J. Burke, N. Gidopoulos, P. Mori-Sánchez, J. P. Perdew, A. Savin, I. Schweigert, S. B. Trickey, W. Yang. Open Problems in Theory: R. J. Bartlett, E. J. Brändas, W. Kohn, J. Ladik, M. Levy, I. Lindgren. We gratefully acknowledge support for the Congress from the following sources: Elsevier; The Donors of the American Chemical Society Petroleum Research Fund; U. S. Army Research Office; Weill Medical College of Cornell University and Intelliome Advanced Sciences LLC. We also greatly appreciate the fine work of UNO Conference Services, which handled all administrative aspects of the Congress. We specifically thank Anne O'Heren Jakob, Director; Patricia Arteaga, Coordinator; and Maria Noveva and Gary Cessna, Program Specialists.