On Special Properties for Continuous Convex Operators and Related Linear Operators

One of the principal results of the paper is that each scalar-type spectral operator in the quasicomplete locally convex space X is reflexive.The paper also studies in detail the relation between the theory of equicontinuous spectral measures in locally convex spaces and the order properties of equicontinuous Bade complete Boolean algebras of projections.o.Introduction.One of the principal results of this paper is that each scalar-type spectral operator T in the quasicomplete, locally convex space X is reflexive, i.e. the strongly closed subalgebra generated by the identity and T in .P(X), the space of continuous linear operators on X, consists precisely of those continuous linear operators on X which leave invariant each (closed) T-invariant subspace of X.For the case that X is a Banach space, this result was established by Gillespie [9] via an interesting factorization theorem in Banach function spaces, a method which does not appear to extend readily to the more general setting.The present approach, however, avoids factorization theorems by showing directly that each continuous linear functional on the strongly closed algebra generated by a Bade-complete, equicontinuous Boolean algebra of projections in X has a representation of the form < .x, x') for some x E X and x' E X', where X' denotes the dual space of X, a result which goes back to R. Pallu de la Barriere [20] for the case of Abelian von Neumann algebras in Hilbert space.Our method is based on ideas from the theory of Riesz spaces and yields considerable simplification of technique, even in the setting of Banach spaces.The cornerstone of the present paper is the extension of the reflexivity theorem of Bade [2] to the setting of locally convex spaces proved in [6] via the theory of closed spectral measures and further refined and sharpened in [5] using purely intrinsic methods, based on order considerations.One of the new features which emerged from the approach of [5] was a type of "automatic continuity" theorem for a certain class of everywhere defined linear operators, even in the absence of a suitable closed-graph theorem.This idea is exploited in 1 to show that an everywhere

In present paper we prove an existence and give a moments estimate for the local time of Gaussian integrators. Every Gaussian integrator is associated with a continuous linear operator in the space of square integrable functions via white noise representation. Hence, all properties of such process are completely characterized by properties of the corresponding operator. We describe the sufficient conditions on continuous linear non-invertible operator which allow the local time of the integrator to exists at any real point. Moments estimate for local time is obtained. A continuous dependence of local time of Gaussian integrators on generating them operators is established. The received statement improves our result presented in [1].

In this paper, we study the boundedness of global continuous linear operators on smooth manifolds. Using the notion of a global symbol, we extend a classical condition of Hörmander type to guarantee the Lp-Lq-boundedness of global operators. Our approach links the mapping properties of continuous linear operators on smooth manifolds with the Lp-estimates of eigenfunctions of operators including a variety of examples, harmonic oscillators, anharmonic oscillators, etc. First, we investigate Lp-boundedness of pseudo-multipliers in the setting of Hörmander–Mihlin type conditions. We also prove L∞-BMO estimates for pseudo-multipliers. Later, we concentrate our investigation to settle Lp-Lq boundedness of the Fourier multipliers and pseudo-multipliers operators for the range 1<p≤2≤q<∞. On the way to achieve our goal of Lp-Lq boundedness, we prove two classical inequalities, namely, Paley inequality and Hausdorff–Young–Paley inequality for smooth manifolds. Finally, we present some examples about the well-posedness of abstract non-linear equations.

Certain norm related functions of linear operators are considered in the very general setting of not necessarily continuous linear operators in normed spaces. These are shown to be closely related to the theory of precompact, strictly cosingular and a class of Φ− type operators in addition to having applications to perturbation theory. We also obtain some basic continuity and precompactness properties of linear operators in normed spaces which are expressed in terms of the functions under consideration.

In this chapter, we are going to study linear operators on finite-dimensional inner product spaces. In the last chapter, we introduced adjoints of linear maps between possibly different inner product spaces. Here we shall see how the adjoint can be used to understand linear operators on a fixed inner product space. The important operators we study here are the self-adjoint, skew-adjoint, normal, orthogonal, and unitary operators. We shall spend several sections on the existence of eigenvalues, diagonalizability, and canonical forms for these special but important linear operators. Having done that, we go back to the study of general linear maps and operators and establish the singular value and polar decompositions. We also show Schur’s theorem to the effect that complex linear operators have upper triangular matrix representations. This result does not depend on the spectral theorem. It is also possible to give a quick proof of the spectral theorem using only the material covered in Sect. 4.1. The chapter finishes with a section on quadratic forms and how they tie in with the theory of self-adjoint operators.

THE ESSENTIAL SPECTRA OF OPERATORS QUASISIMILAR TO SUBNORMAL OPERATORS

In this paper, we examine a sequence of uncountable iterated function systems (U.I.F.S.), where each term in the sequence is constructed from an uncountable collection of contraction mappings along with a linear and continuous operator. Each U.I.F.S. within the sequence is associated with an attractor, which represents a set towards which the system evolves over time, a Markov-type operator that governs the probabilistic behavior of the system, and a fractal measure that describes the geometric and measure-theoretic properties of the attractor. Our study is centered on analyzing the convergence properties of these systems. Specifically, we investigate how the attractors and fractal measures of successive U.I.F.S. in the sequence approach their respective limits. By understanding the convergence behavior, we aim to provide insights into the stability and long-term behavior of such complex systems. This study contributes to the broader field of dynamical systems and fractal geometry by offering new perspectives on how uncountable iterated function systems evolve and stabilize. In this paper, we undertake a comprehensive examination of a sequence of uncountable iterated function systems (U.I.F.S.), each constructed from an uncountable collection of contraction mappings in conjunction with a linear and continuous operator. These systems are integral to our study as they encapsulate complex dynamical behaviors through their association with attractors, which represent sets towards which the system evolves over time. Each U.I.F.S. within the sequence is governed by a Markov-type operator that dictates its probabilistic behavior and is described by a fractal measure that captures the geometric and measure-theoretic properties of the attractor. The core contributions of our work are presented in the form of several theorems. These theorems tackle key problems and provide novel insights into the study of measures and their properties in Hilbert spaces. The results contribute to advancing the understanding of convergence behaviors, the interaction of Dirac measures, and the applications of Monge–Kantorovich norms. These theorems hold significant potential applications across various domains of functional analysis and measure theory. By establishing new results and proving critical properties, our work extends existing frameworks and opens new avenues for future research. This paper contributes to the broader field of mathematical analysis by offering new perspectives on how uncountable iterated function systems evolve and stabilize. Our findings provide a foundational understanding that can be applied to a wide range of mathematical and real-world problems. By highlighting the interplay between measure theory and functional analysis, our work paves the way for further exploration and discovery in these areas, thereby enriching the theoretical landscape and practical applications of these mathematical concepts.

The function , where is Lerch's transcendent, satisfies the following two-dimensional formally self- adjoint second-order hyperbolic differential equation: where . The corresponding differential expression determines a densely defined symmetric operator (the minimal operator) on the Hilbert space , where . We obtain a description of the domains of definition of some symmetric extensions of the minimal operator. We show that formal solutions of the eigenvalue problem for these symmetric extensions are represented by functional series whose structure resembles that of the Fourier series of . We discuss sufficient conditions for these formal solutions to be eigenfunctions of the resulting symmetric differential operators. We also demonstrate a close relationship between the spectral properties of these symmetric differential operators and the distribution of the zeros of some special analytic functions analogous to the Riemann zeta function. Bibliography: 15 titles.

: The paper shows that several well-known results about continuous linear operators on Banach spaces can be generalized to the wider class of convex processes, as defined by Rockafellar. In particular, the open mapping theorem and the standard bound for the norm of the inverse of a perturbed linear operator can be extended to convex processes. In the last part of the paper, these theorems are exploited to prove results about the stability of solution sets of certain operator inequalities and equations in Banach spaces. These results yield quantitative bounds for the displacement of the solution sets under perturbations in the operators and/or in the right-hand sides. They generalize the standard results on stability of unique solutions of linear operator equations.

In this paper, we continue investigation of the lateral order on vector lattices started in [25]. We consider the complexification of a real vector lattice E and introduce the lateral order on . Our first main result asserts that the set of all fragments of an element of the complexification of an uniformly complete vector lattice E is a Boolean algebra. Then, we study narrow operators defined on the complexification of a vector lattice E, extending the results of articles [22, 27, 28] to the setting of operators defined on complex vector lattices. We prove that every order‐to‐norm continuous linear operator from the complexification of an atomless Dedekind complete vector lattice E to a finite‐dimensional Banach space X is strictly narrow. Then, we prove that every C‐compact order‐to‐norm continuous linear operator from to a Banach space X is narrow. We also show that every regular order‐no‐norm continuous linear operator from to a complex Banach lattice is narrow. Finally, in the last part of the paper we investigate narrow operators taking values in symmetric ideals of compact operators.

In this chapter we study endomorphisms, or linear operators, on semimodules of functions that range in idempotent semirings. Here the specific nature of idempotent analysis exhibits itself in the fact that each linear operator on such a semimodule is an integral operator, that is, has the form $$ \left( {Bh} \right)\left( x \right) = \int {^ \oplus b\left( {x,y} \right) \odot h\left( y \right)d\mu \left( y \right) = \mathop {\inf }\limits_y \left( {b\left( {x,y} \right) \odot h\left( y \right)} \right)} $$ for some idempotent integral kernel b(x,y). In §2.1 we give necessary and sufficient conditions for this function to specify a continuous operator. We give a characterization of weak and strong convergence of operator families in terms of kernels and then, in §2.2, we describe two important operator classes— invertible and compact operators—in the same terms. Here another specific feature of the semialgebra of idempotent linear operators is important—the supply of invertible operators is very small; namely, the group of invertible operators is generated by the diagonal operators and by the homomorphisms of the base. Hence, this group consists of idempotent analogs of weighted translation operators. It follows that all automorphisms of the operator semialgebra are inner automorphisms.

The space $${\cal H}{\cal K}$$ of Henstock-Kurzweil integrable functions on [a, b] is the uncountable union of Frechet spaces $${\cal H}{\cal K}$$ (X). In this paper, on each Frechet space $${\cal H}{\cal K}$$ (X), an F-norm is defined for a continuous linear operator. Hence, many important results in functional analysis, like the Banach-Steinhaus theorem, the open mapping theorem and the closed graph theorem, hold for the $${\cal H}{\cal K}$$ (X) space. It is known that every control-convergent sequence in the $${\cal H}{\cal K}$$ space always belongs to a $${\cal H}{\cal K}$$ (X) space for some X. We illustrate how to apply results for Frechet spaces $${\cal H}{\cal K}$$ (X) to control-convergent sequences in the $${\cal H}{\cal K}$$ space. Examples of compact linear operators are given. Existence of solutions to linear and Hammerstein integral equations is proved.

In an earlier paper, we began a study of linear random operators which have a certain probability of behaving as continuous operators. In this paper we study the pointwise limit in probability of a sequence of such operators, extending the Banach-Steinhaus theorem in a stochastical sense.

This paper is a survey of the results in Aguayo et al. (J Math Phys 54(2), 2013; Indag Math (N.S.) 26(1):191–205, 2015; p-Adic Num Ultrametr Anal Appl 9(2):122–137, 2017) but generalized to the case when the complex Levi-Civita field \(\mathcal {C}\) is replaced by a Hahn field \(\mathbb {K}((G))\) for particular choices of the field \(\mathbb {K}\) and the abelian group G. In particular, we consider the Banach space of countable type c0 consisting of all null sequences of \(\mathbb {K}((G))\), equipped with the supremum norm ∥⋅∥∞ and we define a natural inner product on c0 which induces the norm of c0. Then we present characterizations of normal projections and of compact and self-adjoint operators on c0. As a new result in this paper, we apply the Hahn–Banach theorem to show the existence of the dual operator of a given continuous linear operator on c0 and to show that the dual operator and the adjoint operator coincide.

Altman [1] showed that Riesz-Schauder theory remains valid for a completely continuous linear operator on a locally convex Hausdorflf topological vector space over the complex field. In a later paper [2], he proved an analogue of the Aronszajn-Smith result; specifically, he showed that such an operator possesses a proper closed invariant subspace. The purpose of this paper is to show that Ringrose's theory of superdiagonal forms for compact linear operators [3] can be generalized to the case of a completely continuous linear operator on a locally convex Hausdorff topological vector space over the complex field. However, the proof given in [3] requires considerable modification.