Lower bounds for matrices and its applications in frame theory

Abstract

Lower bounds of non-negative triangular matrices and generalized Hausdorff matrices on the Taylor sequence space are considered. Some estimates are found for their lower bounds that depend on the ℓ1-norm of the columns of the Taylor matrix. Further, we show that similar estimates are obtained if we consider such matrices on the domain space of an arbitrary summability matrix Λ in ℓp, in which again the ℓ1-norm of the columns of the matrix Λ appears. As an application of such estimates to frame theory, we present the concept of Λ-frames for a separable Hilbert space H, as a special case of E-frames which were recently introduced in Talebi and Dehghan (2015). We study some properties of Λ-frames and specially Taylor frames. We characterize all Taylor orthonormal bases, Taylor Riesz bases and Taylor frames starting with an arbitrary orthonormal basis for H. Finally, we characterize all dual Taylor frames for a given Taylor frame.