Vol. 40, No. 2, 1972

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 294: 1
Vol. 293: 1  2
Vol. 292: 1  2
Vol. 291: 1  2
Vol. 290: 1  2
Vol. 289: 1  2
Vol. 288: 1  2
Vol. 287: 1  2
Online Archive
Volume:
Issue:
     
The Journal
Subscriptions
Editorial Board
Officers
Special Issues
Submission Guidelines
Submission Form
Contacts
Author Index
To Appear
 
ISSN: 0030-8730
Matrix inequalities and kernels of linear transformations

Peter Botta

Vol. 40 (1972), No. 2, 285–289
Abstract

Let V be a finite dimensional unitary space and mV the unitary space of m-contravariant tensors based on V with the inner product induced from V . If T is a linear transformation on mV to itself and X = (αji,xj) any positive semidefinite hermitian matrix define

dT(X ) = ∥T(x1 ⊗⋅⋅⋅⊗ xm )∥2

Let  1 be any norm on the space of m × m complex matrices, and 𝒯 = {x1 xm : xi V }. The main result is that if T and S are any two linear transformations on nV to itself then the following are equivalent:

(a) ker(T) ∩𝒯 ⊆ ker(S) ∩𝒯

(b) If X is positive semidefinite hermitian and dT(X) = 0 then ds(X) = 0.

(c) There exists a positive integer k and a constant c > 0 such that for all positive semidefinite hermitian matrices X

c∥X∥m1 (k−1)dT(X) ≧ (ds(X ))k.

Some applications to inequalities for generalized matrix functions are given.

Mathematical Subject Classification 2000
Primary: 15A45
Milestones
Received: 10 November 1970
Published: 1 February 1972
Authors
Peter Botta