Towards a model theory for 2-hyponormal operators.
Let \(H\) be a complex Hilbert space and \(L(H)\) the algebra of all bounded linear operators on \(H\). An operator \(T\in L(H)\) is subnormal if there exist operators \(A\) and \(B\) such that \(\widehat T= \left(\begin{smallmatrix} T & A\\ 0 & B\end{smallmatrix}\right)\) is normal, i.e., \[ [T^*, T]:= T^* T- TT^*= AA^*,\tag{1} \] \[ A^* T= BA^*,\tag{2} \] and \[ [B^*, B]+ A^* A= 0.\tag{3} \] In this paper, the authors define a weakly subnormal operator as follows: an operator \(T\in L(H)\) is weakly subnormal if there exist operators \(A\) and \(B\) such that (1) and (2) hold. Here \(\widehat T\) is called a partially normal extension of \(T\). Example 5.1 shows that the class of weakly subnormal operators is larger than the one of subnormal operators. An operator \(T\) is \(k\)-hyponormal if \(k\times k\) operator matrix \(([T^{*j}, T^i])^k_{i,j=1}\) is positive. The present article characterizes weak subnormality for weighted shifts \(W_\alpha\) defined by \(W_\alpha(e_j)= \alpha_j e_j\), where \(\alpha= \{\alpha_j\}^\infty_{j=0}\) is a weight sequence and \(\{e_j\}^\infty_{j=0}\) is an orthonormal basis for \(H\), and establishes that \(2\)-hyponormal weighted shifts \(W_\alpha\) are weakly subnormal, \(\sigma(\widehat W_\alpha)= \sigma(W_\alpha)\), where \(\sigma(T)\) denotes the spectrum of \(T\), and \(\|\widehat W_\alpha\|=\| W_\alpha\|\). Also, the authors provide an example of a weakly subnormal weighted shift which is not \(2\)-hyponormal.
scientific article; zbMATH DE number 1890568
scientific article