The Fejér-Riesz theorem has inspired numerous generalizations in one and several variables, and for matrixand operator-valued functions. This paper is a survey of some old and recent topics that center around Rosenblum’s operator generalization of the classical Fejér-Riesz theorem. Mathematics Subject Classification (2000). Primary 47A68; Secondary 60G25, 47A56, 47B35, 42A05, 32A70, 30E99.

For p-subordinate perturbations of unbounded normal operators, the change of the spectrum is studied and spectral criteria for the existence of a Riesz basis with parentheses of root vectors are established. A Riesz basis without parentheses is obtained under an additional a priori assumption on the spectrum of the perturbed operator. The results are applied to two classes of block operator mat...

In this paper we develop a natural generalization of Schauder basis theory, we term operator-valued basis or simply ov-basis theory, using operator-algebraic methods. We prove several results for ov-basis concerning duality, orthogonality, biorthogonality and minimality. We prove that the operators of a dual ov-basis are continuous. We also dene the concepts of Bessel, Hilbert ov-basis and obta...

In this paper, we investigate duality of modular g-Riesz bases and g-Riesz bases in Hilbert C*-modules. First we give some characterization of g-Riesz bases in Hilbert C*modules, by using properties of operator theory. Next, we characterize the duals of a given g-Riesz basis in Hilbert C*-module. In addition, we obtain sufficient and necessary condition for a dual of a g-Riesz basis to be again...

in this paper we develop a natural generalization of schauder basis theory, we term operator-valued basis or simply ov-basis theory, using operator-algebraic methods. we prove several results for ov-basis concerning duality, orthogonality, biorthogonality and minimality. we prove that the operators of a dual ov-basis are continuous. we also dene the concepts of bessel, hilbert ov-basis and obt...

For f ∈ S(R), we consider the Bochner-Riesz operator R of index δ > 0 defined by R̂δf(ξ) = (1− |ξ|)+ f̂(ξ). Then we prove the Bochner-Riesz conjecture which states that if δ > max{d|1/p − 1/2| − 1/2, 0} and p > 1 then R is a bounded operator from L(R) into L(R); moreover, if δ(p) = d(1/p − 1/2) − 1/2 and 1 < p < 2d/(d + 1), then Rδ(p) is a bounded operator from L(R) into L(R).

The Riesz basis property of the generalized eigenvector system of a Timoshenko beam with boundary feedback controls applied to two ends is studied in the present paper. The spectral property of the operator A determined by the closed loop system is investigated. It is shown that operator A has compact resolvent and generates a C0 semigroup, and its spectrum consists of two branches and has two ...

G-Frames in Hilbert spaces are a redundant set of operators which yield a representation for each vector in the space. In this paper we investigate the connection between g-frames, g-orthonormal bases and g-Riesz bases. We show that a family of bounded operators is a g-Bessel sequences if and only if the Gram matrix associated to its denes a bounded operator.

We investigate Riesz bases of wavelets generated from multiresolution analysis. This investigation leads us to a study of refinement equations with masks being exponentially decaying sequences. In order to study such refinement equations we introduce the cascade operator and the transition operator. It turns out that the transition operator associated with an exponentially decaying mask is a co...

Abstract We provide the conditions for boundedness of Bochner–Riesz operator acting between two different Grand Lebesgue Spaces. Moreover we obtain a lower estimate constant appearing in Lebesgue–Riesz norm estimation and investigate convergence approximation spaces.