Well-bounded Operators Research Articles

Sequences of the Projection-valued Measures and Functional Calculi

Let a pair be functional calculus, where is a homomorphism from the space of the measurable functions on into the space of all linear bounded operators on a reflexive Banach space . We define a norm of functional calculus by , the convergence of the sequence of functional calculi is a convergence relative to this norm. We study the correspondence between sequences of spectral decompositions, well-bounded operators defined on the reflexive Banach space , and their correspondence with the theory of functional calculus for such operators. In this article, we establish that if a sequence of the projection-valued measures strongly converges to then the sequence of the functional calculi converges to the functional calculus Results of the article can be employed in the modern extensions of the quantum theory and theory of quantum information.

The Banach Algebras $$AC(\sigma )$$ and $$BV(\sigma )$$

The spaces \(BV(\sigma )\) and \(AC(\sigma )\) were introduced as part of a program to find a general theory which covers both well-bounded operators and trigonometrically well-bounded operators acting on a Banach space. Since their initial appearance it has become clear that the definitions could be simplified somewhat. In this paper we give a self-contained exposition of the main properties of these spaces using this simplified approach.

Measurable functional calculi and spectral theory in the reflexive Banach spaces

In this paper, we consider the spectral theory. More precisely, we study the spectral families and their corresponding operators on reflexive Banach spaces. Assume [Formula: see text] is a well-bounded operator on reflexive Lebesgue spaces then the operator [Formula: see text] is a scalar-type spectral operator. We establish that if a weak spectral family [Formula: see text] is concentrated on [Formula: see text] then there exists a linear well-bounded operator [Formula: see text] on the reflexive Banach space [Formula: see text], such that [Formula: see text] holds for all [Formula: see text] and [Formula: see text]. We prove that by assuming [Formula: see text] is a functional calculus on the measurable space [Formula: see text], then there are a semi-finite measure space [Formula: see text] and solitary operator [Formula: see text], and an injective pointwise continuous *-homomorphism [Formula: see text], such that [Formula: see text] holds for any operator of the multiplication [Formula: see text] that multiplies by function [Formula: see text].

Measurable Functional Calculus and Spectral theory

In this article, the spectral theory is considered, we study the spectral families and their correspondence to the operators on the reflexive Banach spaces; assume is a well-bounded operator on reflexive Lebesgue spaces then the operator A is a scalar type spectral operator. It is proven that if a weak spectral family is concentrated on [a,b] then there is a linear well-bounded operator on the reflexive Banach space X such that holds for all and .

$$A_{p}$$ A p weights and strong convergence of operator-valued Fourier series for Stieltjes convolutions of Marcinkiewicz functions

Suppose that \(1<p<\infty \) and let w be a bilateral weight sequence satisfying the discrete Muckenhoupt \(A_{p}\) weight condition (in symbols, \(w\in A_{p}\left( \mathbb {Z}\right) \)). In this setting, the left bilateral shift \(\mathcal {L}\) acting on the sequence space \(\ell ^{p}\left( w\right) \) is known to be a trigonometrically well-bounded operator (that is, \(\mathcal {L}\) has a “unitary-like” spectral decomposition \(E:\mathbb {R}\rightarrow \mathfrak {B}\left( \ell ^{p}\left( w\right) \right) \) that consists of idempotent operators and has additional properties reminiscent of, but weaker than, those that would be inherited from a countably additive Borel spectral measure on \(\mathbb {R}\)). In this framework we show that for any Marcinkiewicz multiplier \(\psi :\mathbb {T} \rightarrow \mathbb {C}\), the operator-valued Fourier series of the Stieltjes convolution \(\psi \,\,\mathfrak {\circledast }\)dE (which can be regarded as an operator-theoretic analogue of the Fourier series of \(\psi \)) converges to \(\psi \,\,\mathfrak {\circledast }\)dE, relative to the strong operator topology of \(\mathfrak {B}\left( \ell ^{p}\left( w\right) \right) \), at each point of \(\mathbb {T}\). In particular, the partial sums of the Fourier series of \(\psi \) are uniformly bounded in the Banach algebra of Fourier multipliers for \(\ell ^{p}\left( w\right) \). This outcome is made possible by the mutually advantageous interplay between spectral theory and real variable methods in harmonic analysis—the fundamental links being the operator ergodic discrete Hilbert transform and exrapolation of weights. These results are transferred to the framework of invertible, disjoint, modulus mean-bounded operators acting on \(L^{p}\) spaces of sigma-finite measures. We note that in a still wider framework than that dealt with here—specifically the class of all super-reflexive Banach spaces—trigonometrically well-bounded operators and their spectral families of projections have recently been shown (Berkson, Studia Math 200:221–246, 2010) to constitute a disguised generalization of the dual conjugates of the operator ergodic discrete Hilbert transform generated by not necessarily power-bounded operators, in particular providing new transference vistas.

Fourier analysis methods in operator ergodic theory on super-reflexive Banach spaces

On reflexive spaces trigonometrically well-bounded operators (abbreviated twbo's'') have an operator-ergodic-theory characterization as the invertible operators $U$ whose rotates transfer'' the discrete Hilbert averages $(C,1)$-boundedly. Twbo's permeate many settings of modern analysis, and this note treats advances in their spectral theory, Fourier analysis, and operator ergodic theory made possible by applying classical analysis techniques pioneered by Hardy-Littlewood and L.C. Young to the R.C. James inequalities for super-reflexive spaces. When the James inequalities are combined with spectral integration methods and Young-Stieltjes integration for the spaces $V_{p}(\mathbb{T}) $ of functions having bounded $p$-variation, it transpires that every twbo on a super-reflexive space $X$ has a norm-continuous $V_{p}(\mathbb{T}) $-functional calculus for a range of values of $p>1$, and we investigate the ways this outcome logically simplifies and simultaneously advances the structure theory of twbo's on $X$. In particular, on a super-reflexive space $X$ (but not on the general reflexive space) Tauberian-type theorems emerge which improve to their $(C,0) $ counterparts the $(C,1) $ averaging and convergence associated with twbo's.

THE SPECTRAL TYPE OF SUMS OF OPERATORS ON NON-HILBERTIAN BANACH LATTICES

AbstractT. A. Gillespie showed that on a Hilbert space the sum of a well-bounded operator and a commuting real scalar-type spectral operator is well-bounded. A longstanding question asked whether this might still hold true for operators on Lp spaces for $1\less p\less \infty $. We show here that this conjecture is false. Indeed for a large class of reflexive spaces, the above property characterizes Hilbert space.

A COMPARISON OF ALGEBRAS OF FUNCTIONS OF BOUNDED VARIATION

AbstractMotivated by problems in the spectral theory of linear operators, we previously introduced a new concept of variation for functions defined on a non-empty compact subset of the plane. In this paper, we examine the algebras of functions of bounded variation one obtains from these new definitions for the case where the underlying compact set is either a rectangle or the unit circle, and compare these algebras with those previously used by Berkson and Gillespie in their theories of AC-operators and trigonometrically well-bounded operators.

A Tauberian Theorem for Ergodic Averages, Spectral Decomposability, and the Dominated Ergodic Estimate for Positive Invertible Operators

Suppose that (Ω,μ) is a σ-finite measure space, and 1 < p < ∞. Let T:Lp(μ → L p(μ) be a bounded invertible linear operator such that T and T−1 are positive. Denote by \({\mathfrak{E}}\)n(T) the nth two-sided ergodic average of T, taken in the form of the nth (C,1) mean of the sequence {Tj+T−j}j =1∞. Martin-Reyes and de la Torre have shown that the existence of a maximal ergodic estimate for T is characterized by either of the following two conditions: (a) the strong convergence of En(T)n=1∞; (b) a uniform App estimate in terms of discrete weights generated by the pointwise action on Ω of certain measurable functions canonically associated with T. We show that strong convergence of the (C,2) means of {Tj+T−j}j=1∞ already implies (b). For this purpose the (C,2) means are used to set up an `averaged' variant of the requisite uniform Ap weight estimates in (b). This result, which can be viewed as a Tauberian-Type replacement of (C,1) means by (C,2) means in (a), leads to a spectral-theoretic characterization of the maximal ergodic estimate by application of a recent result of the authors establishing the strong convergence of the (C,2)-weighted ergodic means for all trigonometrically well-bounded operators. This application also serves to equate uniform boundedness of the rotated Hilbert averages of T with the uniform boundedness of the ergodic averages En(T)n = 1∞.

On some classes of AC-operators

AC-operators generalise normal operators on Hilbert space in the context of well-boundedness. In this paper we study AC-operators T ¼ U þ iV; where U and V are commuting well-bounded operators with decomposition of the identity of bounded variation. We also explore some properties of AC-operators by applying the theory of (Foiac¸ c s) decomposable operators.

WELL-BOUNDED OPERATORS ON NON-REFLEXIVE BANACH SPACES II

In this paper we examine the relationship between the various subclasses of well-bounded operators. Although some of these subclasses have been shown to coincide on certain families of Banach spaces, we show here that apart from these previously known cases, the subclasses are typically distinct.

Compact well-bounded operators

Every compact well-bounded operator has a representation as a linear combination of disjoint projections reminiscent of the representation of compact self-adjoint operators. In this note we show that the converse of this result holds, thus characterizing compact well-bounded operators. We also apply this result to study compact well-bounded operators on some special classes of Banach spaces such as hereditarily indecomposable spaces and certain spaces constructed by G. Pisier.

Spectral decompositions, ergodic averages, and the Hilbert transform

Let U be a trigonometrically well-bounded operator on a Banach space X, and denote byfAn(U)g 1=1 the sequence of (C; 2) weighted discrete ergodic averages of U, that is, An(U) = 1 n X 0<jkj n 1 jkj n + 1 U k : We show that this sequence fAn(U)g 1=1 of weighted ergodic averages converges in the strong operator topology to an idempotent operator whose range isfx2 X : Ux = xg; and whose null space is the closure of (I U)X. This result expands the scope of the traditional Ergodic Theorem, and thereby serves as a link between Banach space spectral theory and ergodic operator theory. We also develop a characterization of trigonometrically well-bounded operators by their ability to \transfer the discrete Hilbert transform to the Banach space setting via (C; 1) weighting of Hilbert averages, and these results together with those on weighted ergodic averages furnish an explicit expression for the spectral decomposition of a trigonometrically well-bounded operatorU on a Banach space in terms of strong limits of appropriate averages of the powers of U. We also treat the special circumstances where corresponding results can be obtained with the (C; 1) and (C; 2) weights removed.

Well-bounded operators on general Banach spaces

An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Well-bounded operators on nonreflexive Banach spaces

Every well-bounded operator on a reflexive Banach space is of type (B), and hence has a nice integral representation with respect to a spectral family of projections. A longstanding open question in the theory of well-bounded operators is whether there are any nonreflexive Banach spaces with this property. In this paper we extend the known results to show that on a very large class of nonreflexive spaces, one can always find a well-bounded operator which is not of type (B). We also prove that on any Banach space, compact well-bounded operators have a simple representation as a combination of disjoint projections.

The spectral theorem for well-bounded operators

AbstractWell-bounded operators are those which possess a bounded functional calculus for the absolutely continuous functions on some compact interval. Depending on the weak compactness of this functional calculus, one obtains one of two types of spectral theorem for these operators. A method is given which enables one to obtain both spectral theorems by simply changing the topology used. Even for the case of well-bounded operators of type (B), the proof given is more elementary than that previously in the literature.

Unbounded well-bounded operators, strongly continuous semigroups and the Laplace transform

Suppose A is a (possibly unbounded) linear operator on a Banach space. We show that the following are equivalent. (1) A is well-bounded on [0,∞). (2) -A generates a strongly continuous semigroup ${e^{-sA}}_{s≤0}$ such that ${(1/s^2)e^{-sA}}_{s>0}$ is the

Well-Bounded and Scalar-Type Spectral Operators on Lp Spaces

Well-Bounded and Scalar-Type Spectral Operators on <i>L^p </i> Spaces

Well-Bounded and Scalar-Type Spectral Operators on Spaces not Containing C 0

Well-Bounded and Scalar-Type Spectral Operators on Spaces not Containing C 0

Well-bounded and scalar-type spectral operators on spaces not containing 𝑐₀

It is known that a necessary and sufficient condition for a well-bounded operator on a weakly complete complex Banach space to be scalar-type spectral is that its decomposition of the identity be of bounded variation. We show in this paper that this condition is necessary and sufficient exactly when the Banach space does not contain a subspace isomorphic to c 0 {c_0} .