Positive transformations restricted to subspaces and inequalities among their proper values

Abstract:

Let A be a positive Hermitian transformation on an n-dimensional unitary space ${E_n}$ with proper values ${a_1} \geqq \cdots \geqq {a_n}$. Let ${b_1} \geqq \cdots \geqq {b_k}$ be the proper values of $A|M$, where M is a proper subspace of ${E_n}$ and ${c_1} \geqq \cdots \geqq {c_h}$ be the proper values of $A|{M^ \bot }$. Let ${i_1} < \cdots < {i_r}$ and ${j_1} < \cdots < {j_r}$ be sequences of positive integers, with ${i_r} \leqq k$ and ${j_r} \leqq h$. Then $({b_{{i_1}}} \cdots {b_{{i_r}}}) \cdot ({c_{{j_1}}} \cdots {c_{{j_r}}}) \geqq ({a_{n - r + 1}} \cdots {a_n})({a_{{i_1} + {j_1} - 1}} \cdots {a_{{i_r} + {j_r} - 1}})$. In this article generalizations of this inequality have been studied.