## SOME NEW APPROACHES ON PROPERTIES OF U-CROSS GRAM MATRIX

## SOME NEW APPROACHES ON PROPERTIES OF U-CROSS GRAM MATRIX

*R. Mehrdad, E. Osgooei*

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## Abstract

The operator equation Uf = b can be transferred to matrix level by Mc = d, where M can be considered as a generalization of Gram matrix. This matrix obtained by the composition of the analysis and synthesis operators of two different sequences with inserting an operator U on a Hilbert space, called a U-cross Gram matrix. In this paper, we investigate the U-cross Gram operator GU,Φ,Ψ, associated to the sequences {φk}∞k=1 and {ψk}∞k=1 and sufficient and necessary conditions for boundedness, invertibility, com- pactness of this operator are determined depending on the associated sequences. We show that invertibility of GU,Φ,Ψ is not possible when the associated sequences are frames but not Riesz Bases or at most one of them is a Riesz basis.

## Keywords

U-cross Gram matrix, Riesz basis, Invertible operator, Hilbert space.