m-Isometric Operators with Null Symbol and Elementary Operator Entries - Bhagwati Prashad Duggal (Axioms)
A pair <semantics>(A,B)</semantics> of Banach space operators is strict <semantics>(m,X)</semantics>-isometric for a Banach space operator <semantics>X∈B(X)</semantics> and a positive integer m if <semantics><msubsup>▵A,Bm</msubsup>(X)=<mfenced separators="" open="(" close=")"><mstyle displaystyle="true"><munderover>∑j=0m</munderover></mstyle><mfenced separators="" open="(" close=")"><mtable><mtr><mtd>m</mtd></mtr><mtr><mtd>j</mtd></mtr></mtable></mfenced><msubsup>LAj</msubsup><msubsup>RBj</msubsup></mfenced>(X)=0</semantics> and <semantics><msubsup>▵A,Bm−1</msubsup>(X)≠0</semantics>, where <semantics>LA</semantics> and <semantics>RB∈B(B(X))</semantics> are, respectively, the operators of left multiplication by A and right multiplication by B. Define operators <semantics>EA,B</semantics> and <semantics>EA,B(X)</semantics> by <semantics>EA,B=LARB</semantics> and <semantics><mfenced separators="" open="(" close=")">EA,B(X)</mfenced>n=<msubsup>EA,Bn</msubsup>(X)</semantics> for all non-negative integers n. Using little more than an algebraic argument, the following generalised version of a result relating <semantics>(m,X)</semantics>-isometric properties of pairs <semantics>(A1,A2)</semantics> and <semantics>(B1,B2)</semantics> to pairs <semantics>(EA1,A2(S1),EB1,B2(S2))</semantics> and <semantics>(EA1,A2,EB1,B2)</semantics> is proved: if <semantics>Ai,Bi,Si,X</semantics> are operators in <semantics>B(X)</semantics>, <semantics>1≤i≤2</semantics> and X a quasi-affinity, then the pair <semantics>(EA1,A2(S1),EB1,B2(S2))</semantics> (resp., the pair <semantics>(EA1,A2,EB1,B2)</semantics>) is strict <semantics>(m,X)</semantics>-isometric for all <semantics>X∈B(X)</semantics> if and only if there exist positive integers <semantics>mi≤m</semantics>, <semantics>1≤i≤2</semantics> and <semantics>m=m1+m2−1</semantics>, and a non-zero scalar <semantics>β</semantics> such that <semantics>I−EβA1,A2(S1)</semantics> is (strict) <semantics>m1</semantics>-nilpotent and <semantics>I−E<mstyle scriptlevel="0" displaystyle="true">1β</mstyle>B1,B2(S2)</semantics> is (strict) <semantics>m2</semantics>-nilpotent (resp., <semantics>(βA1,B1)</semantics> is strict <semantics>(m1,I)</semantics>-isometric and <semantics>(<mstyle scriptlevel="0" displaystyle="true">1β</mstyle>B2,A2)</semantics> is strict <semantics>(m2,I)</semantics>-isometric).