Eigenvalues of Nuclear Operators of Diagonal Type.

It was shown by M.I. Zelikin (2007) that the spectrum of a nuclear operator in a Hilbert space is central-symmetric iff the traces of all odd powers of the operator equal zero. B. Mityagin (2016) generalized Zelikin’s criterium to the case of compact operators (in Banach spaces) some of which powers are nuclear, considering even a notion of so-called \( \mathbb Z_d \)-symmetry of spectra introduced by him. We study α-nuclear operators generated by the tensor elements of so-called α-projective tensor products of Banach spaces, introduced in the paper (α is a quasi-norm). We give exact generalizations of Zelikin’s theorem to the cases of \( \mathbb Z_d \)-symmetry of spectra of α-nuclear operators (in particular, for s-nuclear and for (r, p)-nuclear operators). We show that the results are optimal.

A long-standing problem that arises when constructing the mathematical apparatus of quantum mechanics is the need to work with unbounded operators. Since the space of nuclear operators is preconjugate for the algebra of all bounded operators, we can consider the states of a quantum system as nuclear operators, and consider bounded operators as observables. In this case, taking the trace for the product of a nuclear operator (a quantum state) and a bounded operator (a quantum observable) gives the average value of a quantum observable in a fixed state. The existence of such an average for unbounded operators is not guaranteed. If we want to define a space of observables that includes such naturally occurring unbounded operators as the position and momentum operators, for which average values are always determined, we should consider a space of states smaller than all nuclear operators. Recently, this approach has been mathematically correct implemented in the Hilbert space H = L^2(R^N). The so-called space of Schwarz operators, equipped with a system of semi-norms and being a Frechet space, was chosen as the space of states. Schwarz operators appeared to be integral operators whose kernels are functions belonging to the usual Schwarz space. The dual space to the space of Schwarz operators should be considered as the space of quantum observables and it really includes such standard observables as polynomials from the products of the position and momentum operators. In the present work we transfer this approach to the symmetric Fock space H = F(H) over an infinite dimensional separable Hilbert space H. We introduce the space of Schwartz operators in F(H) over an infinite dimensional separable space H and investigate which of the standard operators of quantum white noise belong to the space dual to the space of Schwartz operators.

Let Σ be a σ-algebra of subsets of a set Ω and $B(\Sigma)$ be the Banach space of all bounded Σ-measurable scalar functions on Ω. Let $\tau(B(\Sigma),ca(\Sigma))$ denote the natural Mackey topology on $B(\Sigma)$. It is shown that a linear operator T from $B(\Sigma)$ to a Banach space E is Bochner representable if and only if T is a nuclear operator between the locally convex space $(B(\Sigma),\tau(B(\Sigma),ca(\Sigma)))$ and the Banach space E. We derive a formula for the trace of a Bochner representable operator $T:B({\cal B} o)\rightarrow B({\cal B} o)$ generated by a function $f\in L^1({\cal B} o, C(\Omega))$, where Ω is a compact Hausdorff space.

Abstract. The necessary and sufficient conditions for stability of abstract difference schemes in Hilbert and Banach spaces are formulated. Contrary to known stability results we give stability conditions for schemes with non-self-adjoint operator coefficients in a Hilbert space and with strongly positive operator coefficients in a Banach space. It is shown that the parameters of the sectorial spectral domain play the crucial role. As an application we consider the Richardson iteration scheme for an operator equation in a Banach space, in particulary the Richardson iteration with precondition for a finite element scheme for a non-selfadjoint operator. The theoretical results are also the basis when using the regularization principle to construct stable difference schemes. For this aim we start from some simple scheme (even unstable) and derive stable schemes by perturbing the initial operator coefficients and by taking into account the stability conditions. Our approach is also valid for schemes with unbounded operator coefficients.

We study lattice summing operators between Banach spaces focusing on two classes, ‐summing and strongly ‐summing operators, which are generated by Orlicz sequence lattices . For the class of strongly ‐summing operators, we prove the domination theorem, which complements Pietsch's fundamental domination theorem for ‐summing operators. Based on this result, we show that strongly ‐summing operators are Dunford–Pettis. As a consequence, we show that these classes are, in general, distinct. We also demonstrate that the class of strongly ‐summing operators between Hilbert spaces coincides with the Hilbert–Schmidt class when is a separable Orlicz space. Finally, we consider generalized nuclear operators, and using a factorization description, we prove that ‐nuclear operators are ‐summing when is separable.

Let X be a completely regular Hausdorff space and E and F be Banach spaces. Let C_{\mathrm{rc}}(X,E) denote the Banach space of all continuous functions f:X\rightarrow E such that f(X) is a relatively compact set in E . Let \beta_{\sigma} be the strict topology on C_{\mathrm{rc}}(X,E) . We characterize the nuclearity of a (\beta_{\sigma}, \|\cdot\|_{F}) -continuous operator T:C_{\mathrm{rc}}(X,E)\rightarrow F in terms of its representing operator-valued Baire measure. As an application, we establish the relationship between the nuclearity of a (\beta_{\sigma},\|\cdot\|_{F}) -continuous operator T:C_{\mathrm{rc}}(X,E)\rightarrow F and the nuclearity of its conjugate operator T' .

Characterizations of the metric and generalized metric projections on subspaces of Banach spaces

Spaces of σ(p)-nuclear linear and multilinear operators on Banach spaces and their duals

A note on nuclear operators in Hilbert space

The lattice of closed ideals in the Banach algebra of operators on certain Banach spaces

Many of the fundamental research problems in the geometry of normed linear spaces can be loosely phrased as: Given a Banach space X and a class of Banach spaces Y does X contain a subspace Y ∈ Y? As a Banach space X is determined by its unit ball B x ≡ { x ∈ X :‖ x ‖ ≤ 1 } the problem can be rephrased in terms of the geometry of convex sets: Can a given unit ball B x be sliced with a subspace to obtain a set in some given class of unit balls? A result of this type is the famous theorem of Dvoretzky (see also [L], [M6], [M4], [MS], [FLM]).

In this paper, apply an established transference principle to obtain the boundedness of certain functional calculi for the sequence of semigroup generators. It is proved that if be the sequence generates 0- semigroups on a Hilbert space, then for each the sequence of operators has bounded calculus for the closed ideal of bounded holomorphic functions on right half–plane. The bounded of this calculus grows at most logarithmically as. As a consequence decay at ∞. Then showed that each sequence of semigroup generator has a so-called (strong) m-bounded calculus for all m∈ℕ, and that this property characterizes the sequence of semigroup generators. Similar results are obtained if the underlying Banach space is a UMD space. Upon restriction to so-called semigroups, the Hilbert space results actually hold in general Banach spaces.

Sequence spaces are one of the topics in the field functional analysis such as null sequence space, converging sequence space and bounded sequence space designated and respectively. Those sequence spaces are normed spaces. Entry of Copson matrices denoted by with value and 0 for other . Null sequence space, , converging sequence space and bounded sequence space with Copson matrices built new sequence spaces that are squence spaces . The purpose of this study is to get some properties that occur to those sequence spaces with the same normed as sequence space . The results of this study are sequence spaces are Banach spaces with mapping coordinate is continues (BK), there are inclusion properties on sequence spaces, sequence space has AK property and sequence spaces and are compact.

This volume provides a self-contained introduction to the theory of tensor products of Banach spaces. It is written for graduate students in analysis or for researchers in other fields who wish to become acquainted with this area. The only prerequisites are a basic knowledge of functional analysis and measure theory. Features of particular interest include: - A full treatment of the Grothendieck theory of tensor norms; - Coverage of the Chevet-Saphar norms and their duals, along with the associated classes of nuclear, integral and summing operators; - Chapters on the approximation property and the Radon-Nikodym property; - Topics such as the Bochner and Pettis integrals, the principle of local reflexivity and the Grothendieck inequality placed in a natural setting; - The classes of operators generated by a tensor norm and connections with the theory of operator ideals. Each chapter is accompanied by worked examples and a set of exercises, and two appendices provide essential material on summability in Banach spaces and properties of spaces of measures that may be new to the beginner.

This chapter develops the beginnings of abstract functional analysis, a subject designed to study properties of functions by treating the functions as the members of a space and formulating the properties as properties of the space. Section 1 defines Banach spaces as complete normed linear spaces and gives a number of examples of these. The space of bounded linear operators from one normed linear space to another is a normed linear space, and it is a Banach space if the range is a Banach space. Sections 2–3 concern Hilbert spaces. These are Banach spaces whose norms are induced by inner products. Section 2 shows that closed vector subspaces of such a space have orthogonal complements, and it shows the role of orthonormal bases for such a space. Section 3 concentrates on bounded linear operators from a Hilbert space to itself and constructs the adjoint of each such operator. Sections 4–6 prove the three main abstract theorems about the norm topology of general normed linear spaces—the Hahn–Banach Theorem, the Uniform Boundedness or Banach–Steinhaus Theorem, and the Interior Mapping Principle. A number of consequences of these theorems are given. The second and third of the theorems require some hypothesis of completeness. The topic of Hilbert and Banach spaces continues in Chapter IV of the companion volume, Advanced Real Analysis.