Geometric Mean for Toeplitz-Block and Toeplitz Lemma in Geodesic Metric Space

Abstract

The effective and computing matrix geometric means satisfying the Ando-Li- Mathias properties are studied. The geometric means of structured matrices and for Toeplitz and Toeplitz -block-block-Toeplitz matrices are also studied. We show the regular operator mappings, monotone and multiplicative geometric means for a class of Toeplitz matrices and the Ka ̈hler mean of block – Toeplitz matrices with Toeplitz structured blocks. We explain and develop Beurling-Malliavin theory and geometric significance for Toeplitz Kernels with multipliers between model spaces and improved Toeplitz order. We determine the spectral gaps for sets, measures, two weight inequality for Hilbert transform, higher order multi-dimensional extensions of Cesa ̀ro theorem, and law- of large numbers for weighted inductive means in a Hadamard space. The Toeplitz lemma in geodesic metric space, convergence in probability, mean convergence, complete convergence with moment and convergence of inductive means are investigated and obtained.