More Inequalities and Asymptotics for Matrix Valued Linear Positive Operators: the Noncommutative Case

In a recent paper, the author and P. Tilli have analyzed Cauchy-Schwarz type inequalities and associated relations in the case of matrix valued linear positive operators acting on some space A of scalar valued functions. Here we deal with the noncommutative case where the space A is constituted by matrix valued functions. Some inequalities are formally the same, but many of them are substantially affected by the noncommutativity of the underlying space. Several applications to Toeplitz sequences with matrix valued symbols are finally derived and discussed.

[1]  Paolo Tilli,et al.  Locally Toeplitz sequences: spectral properties and applications , 1998 .

[2]  Stefano Serra Capizzano,et al.  Spectral and structural analysis of high precision finite difference matrices for elliptic operators , 1999 .

[3]  Paolo Tilli,et al.  A note on the spectral distribution of toeplitz matrices , 1998 .

[4]  Paolo Tilli,et al.  On unitarily invariant norms of matrix-valued linear positive operators , 2002 .

[5]  A. Böttcher,et al.  Toeplitz Matrices, Asymptotic Linear Algebra, and Functional Analysis , 2000 .

[6]  Eugene E. Tyrtyshnikov,et al.  Spectra of multilevel toeplitz matrices: Advanced theory via simple matrix relationships , 1998 .

[7]  R. Bhatia.  Matrix Analysis , 1996 .

[8]  D. Fasino,et al.  Spectral clustering properties of block multilevel Hankel matrices , 2000 .

[9]  Bernd Silbermann,et al.  Analysis of Toeplitz Operators , 1991 .

[10]  Stefano Serra Capizzano,et al.  Spectral Analysis of (Sequences of) Graph Matrices , 2001, SIAM J. Matrix Anal. Appl..

[11]  E. E. Tyrtyshnikov.  Influence of matrix operations on the distribution of Eigenvalues and singular values of Toeplitz matrices , 1994 .

[12]  D. Fasino,et al.  From Toeplitz matrix sequences to zero distribution of orthogonal polynomials , 2001 .

[13]  A. Böttcher,et al.  Introduction to Large Truncated Toeplitz Matrices , 1998 .

[14]  S. Serra,et al.  Spectral and Computational Analysis of Block Toeplitz Matrices Having Nonnegative Definite Matrix-Valued Generating Functions , 1999 .

[15]  Stefano Serra,et al.  Asymptotic Results on the Spectra of Block Toeplitz Preconditioned Matrices , 1999 .

[16]  Stefano Serra Capizzano,et al.  An ergodic theorem for classes of preconditioned matrices , 1998 .

[17]  S. Serra Capizzano,et al.  Distribution results on the algebra generated by Toeplitz sequences: a finite-dimensional approach , 2001 .

[18]  Stefano Serra-Capizzano,et al.  Positive representation formulas for finite difference discretizations of (elliptic) second order PDEs , 1999 .

[19]  H. Widom.  Asymptotic behavior of block Toeplitz matrices and determinants. II , 1974 .

[20]  E. E. Tyrtyshnikov.  A unifying approach to some old and new theorems on distribution and clustering , 1996 .

[21]  Steffen Roch,et al.  C* - Algebras and Numerical Analysis , 2000 .