Matrix inequalities and kernels of linear transformations
Peter Botta · Pacific Journal of Mathematics · 1972
Let V be a finite dimensional unitary space and ® m F the unitary space of m-contravariant tensors based on V with the inner product induced from V. If Γ is a linear transformation on ® m F to itself and X=(Xi,Xj) any positive semidefinite hermitian matrix define Let || ||i be any norm on the space of m X m complex matrices, and J^~= {xi ® * ® 2C»»-Λ?i β V}.The main result is that if T and £ are any two linear transformations on (x) m V to itself then the following are equivalent: (a) ker(T)nicker (S)Π^( b) If X is positive semidefinite hermitian and d τ {X) = 0 then cF(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\\^ι ) d τ {X) ^ (d s (X)) k .Some applications to inequalities for generalized matrix functions are given.1* Introduction* Let V be a finite dimensional unitary space with inner product ( , ) and ® m F the space of m-contravariant tensors based on V.The inner product on V induces an inner product on ® m F as follows.If x u , x m ; y lf •••,!/"€ V define (%ι (X) (X) ««, l/i (8) <g)2/«) = Π (»i, 2/i) i=i