Forms of biisometric operators and biorthogonality

Abstract

The paper proves two results involving a pair ( A , B ) of P-biisometric or ( m , P ) -biisometric Hilbert-space operators for arbitrary positive integer m and positive operator P. It is shown that if A and B are power bounded and the pair ( A , B ) is ( m , P ) -biisometric for some m, then it is a P-biisometric pair. The important case when P is invertible is treated in detail. It is also shown that if ( A , B ) is P-biisometric, then there are biorthogonal sequence with respect to the inner product 〈 ⋅ ; ⋅ 〉 P = 〈 P ⋅ ; ⋅ 〉 that have a shift-like behavior with respect to this inner product.