A random Banach-Steinhaus theorem

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.

A classical result due to M. Eidelheit and B. Yood states that the standard algebra norm on the algebra of bounded linear operators on a Banach space is minimal, in the sense that the norm must be less than a multiple of any other submultiplicative norm on the same algebra. This definition does not assume that the arbitrary algebra norm is complete. In cases when the standard algebra norm is, in addition, maximal, it is therefore unique up to equivalence. More recently, M. Meyer showed that the Calkin algebras of a very restricted class of Banach spaces also have unique algebra norms. We generalise the Eidelheit-Yood method of proof, to show that the conventional quotient norm on a larger class of Calkin algebras is minimal. Since maximality of the norm is a presumed property for the class, the norm is also unique. We thus extend the result of Meyer. In particular, we establish that the Calkin algebras of canonical Banach spaces such as James’ space and Tsirelson’s space have unique algebra norms, without assuming completeness. We also prove uniqueness of norm for quotients of the algebras of operators on classical non-separable spaces, the closed ideals of which were previously studied by M. Daws. One aspect of the Eidelheit-Yood method is a dependence on the uniform boundedness principle. As a component of our generalisation, we prove an analogue of that principle which applies to Calkin algebra elements rather than bounded linear operators. In order to translate the uniform boundedness principle into this new setting, we take the perspective that non-compact operators map certain wellseparated sequences to other well-separated sequences. We analyse the limiting separation of such sequences, using these values to measure the non-compactness of operators and define the requisite notion of a bounded set of non-compact operators. In the cases when the underlying Banach space has a Schauder basis, we are able to restrict attention to seminormalised block basic sequences. As a consequence, our main uniqueness of norm result for Calkin algebras relies on the existence of bounded mappings between, and projections onto, the spans of block basic sequences in the relevant Banach spaces.

This paper provides a uniform boundedness theorem for a class of convex operators, such as Banach–Steinhaus theorem for families of continuous linear operators. The case of continuous symmetric sublinear operators is outlined. Second, a general theorem characterizing the existence of the solution of the Markov moment problem is reviewed, and a related minimization problem is solved. Convexity is the common point of the two aims of the paper mentioned above.

This is a survey (without proofs except for verifying a few new facts) of the slowest possible rate of convergence of a sequence of linear operators that converges pointwise to a linear operator. A sequence of linear operators (L n ) is said to converge to a linear operator Larbitrarily slowly (resp., almost arbitrarily slowly) provided that (L n ) converges to L pointwise, and for each sequence of real numbers (ϕ(n)) converging to 0, there exists a point x = x ϕ such that \(\|{L}_{n}(x) - L(x)\| \geq \phi (n)\) for all n (resp., for infinitely many n). Two main “lethargy” theorems are prominent in this study, and they have numerous applications. The first lethargy theorem (Theorem 11.16) characterizes almost arbitrarily slow convergence. Applications of this lethargy theorem include the fact that a large class of polynomial operators (e.g., Bernstein, Hermite–Fejer, Landau, Fejer, and Jackson operators) all converge almost arbitrarily slowly to the identity operator. Also all the classical quadrature rules (e.g., the composite Trapezoidal Rule, composite Simpson’s Rule, and Gaussian quadrature) converge almost arbitrarily slowly to the integration functional. The second lethargy theorem (Theorem 11.21) gives useful sufficient conditions that guarantee arbitrarily slow convergence. In the particular case when the sequence of linear operators is generated by the powers of a single linear operator, there is a “dichotomy” theorem (Theorem 11.27) which states that either there is linear (fast) convergence or arbitrarily slow convergence; no other type of convergence is possible. Some applications of the dichotomy theorem include generalizations and sharpening of (1) the von Neumann-Halperin cyclic projections theorem, (2) the rate of convergence for intermittently (i.e., “almost” randomly) ordered projections, and (3) a theorem of Xu and Zikatanov.

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.

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.

In this central chapter, the fundamental elements of functional analysis are presented. Although topological vector spaces are considered, essentially for the use of weak topologies, the focus is on normed spaces. Normed spaces in duality or metric duality form a convenient framework. The main pillars of functional analysis are presented: separation properties, the uniform boundedness theorem, the open mapping theorem, the closed graph theorem…Some special properties such as reflexivity, separability, and uniform convexity are given some attention. An account of spectral theory for linear operators is presented and the case of compact operators is considered. The chapter ends with a presentation of regulated functions and functions of bounded variation. This enables us to dispose of an elementary integration theory that is often sufficient for simple purposes.

The algorithm of application of the method of steepest descent to the solution of problems of structural mechanics and solid mechanics, described by nonlinear differential equations. For application of this method to nonlinear operators are described by a sequence of linear operators in incremental form, unlimited and complex linear operator, in line with the idea of L. V. Kantorovich is limited to a simple linear unbounded operator.

The paper centers around a pair of sequences of linear positive operators. The former has the degree of exactness one and the latter has its degree of exactness null, but, as a novelty, it reproduces the third test function of Korovkin theorem. Quantitative estimates of the rate of convergence are given in different function spaces traveling from classical approximation to approximation in abstract spaces. Particular classes are also studied.

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.

Let X and Y be vector lattices and Ti: X → Y a sequence of linear operators which are sequentially continuous with respect to relative uniform convergence. If {Tjx} is relatively uniformly convergent to Tx for each x ∈ X, under appropriate assumptions on the spaces, we show that the linear operator T is also continuous and that the {Ti} are order equicontinuous in a certain sense. We also establish an order version of the Uniform Boundedness Principle.

We define for a linear and continuous operator T between Banach spaces E and F a sequence (Tn(T)) as where the supremum is taken over all normalized elements x1,…. xn of E and all normalized linear functionals b1,…, bn on F. After proving basic properties we give estimates from above by special ideal norms and derive from this related results of eigenvalue distributions.

In this work we consider quantitatively with rates the convergence of sequences of linear operators applied on Banach space valued functions. The results are pointwise estimates with rates. To prove our main results we use an elegant and natural boundedness property of our linear operators by their companion positive linear operators. Our inequalities are generalized g-direct and iterated fractional involving the right and left vector Caputo type generalized g-direct and iterated fractional derivatives, built in vector moduli of continuity. We treat wide and general classes of Banach space valued functions. We give applications to vectorial Bernstein operators. See also [6].

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