Trace-class interpolation for vectors in tridiagonal algebras

Abstract

Given vectors x and y in a Hilbert space, an interpolatingoperator is a bounded operator T such that Tx=y. An interpolatingoperator for n vectors satisfies the equation Tx_i =y_i,for i=1,2,cdots,n.In this article, we obtained the following :Let x=(x_i) and y=(y_i) be two vectors in Cal H such that x_i neq 0for all i=1,2,cdots.Then the following statements are equivalent.itemitem{rm (1)} There exists an operator A in AlgCal L such thatAx =y, A is a trace-class operatorand every E in Cal L reduces A.itemitem{rm (2)} displaystyle sup left{ {{| sum_{k=1}^l alpha_k E_k y|}over {| sum_{k=1}^l alpha_k E_k x|}}: l in N, alpha_k in {Bbb C} text{and} E_k in {Cal L} right} < inftylinebreak and displaystylesum_{n=1}^infty |y_n| |x_n|^{-1} <infty.