Biorthogonal expansions for symmetrizable operators (Q653820)

Given a positive operator \(A\) on a complex Hilbert space \({\mathcal H}\), another bounded linear operator \(B\) on \({\mathcal H}\) is said to be symmetrizable with respect to \(A\) provided that the operator \(AB\) is selfadjoint; that is, provided that \(AB=B^*A\). The main result (Theorem 3.1) of this interesting and readable article is as follows. Suppose that \(B\) is symmetrizable with respect to the positive operator \(A\), the range inclusion \(\text{ran } B^*\subset \text{ran } A^{1/2}\) holds, and the set \(\{A^{1/2}Bx:x\in {\mathcal H},\langle Ax,x\rangle \leq 1\}\) is precompact in \({\mathcal H}\). Let \(P\) denote the orthogonal projection onto the closure of the range of \(A\). Then there exists a sequence of nonzero real numbers \(\{\lambda_k\}_{k\geq 1}\) which is either finite or is such that the sequence \(\{| \lambda_k| \}_{k\geq 1}\) is strictly decreasing with limit 0, and there exists a biorthogonal sequence of vectors \(\{e_n,f_n\}_{n\in\mathbb N}\) such that \[ PBx= \sum_{k=1}^{\infty}{\lambda_k\langle x,\,f_k\rangle e_k}\quad\text{and}\;B^*Px= \sum_{k=1}^{\infty}{\lambda_k\langle x,\,e_k\rangle f_k}\;,\;\forall x\in {\mathcal H}. \] Lemma 2.1 and Proposition 2.2 show that each of the conditions \(\text{ran\,} B^*\subset \text{ran\,} A^{1/2}\) and that the set \(\{A^{1/2}Bx\,:\,x\in {\mathcal H},\,\langle Ax,\,x\rangle\leq 1\}\) be precompact in \({\mathcal H}\) (from the Theorem) is equivalent to several other conditions, any of which, therefore, could be used instead. In particular, the latter condition is equivalent to the compactness of a certain selfadjoint operator \(S\) satisfying the relation \(SA^{1/2}=A^{1/2}B\). The existence of such an \(S\) is due to \textit{J. Dieudonné} [Proc.\ Int.\ Symp.\ Linear Spaces, Jerusalem 1960, 115--122 (1961; Zbl 0114.31601)] and is explained in the present paper through the intermediary of the Hilbert space \(H_A\) arising from the inner product \(\langle Ax,\,Ay\rangle_A:=\langle Ax,\,y\rangle\) defined on the range of \(A\). With the assumption that \(S\) is compact, then, the sequence \(\{\lambda_k\}\) in the main theorem comprises the eigenvalues of \(S\), while the vectors \(\{e_k\}\) are such that \(A^{1/2}e_k\) is an eigenvector of \(S\) corresponding to \(\lambda_k\) for each \(k\). The proof of Theorem 3.1 shows that the sequence \(\{f_k:=Ae_k\}\) both satisfies the biorthogonality requirement that \(\langle f_j,\,e_k\rangle \) is equal to 0 whenever \(j\neq k\) and 1 when \(j=k\) and leads to the result of the theorem. Thus, the spectral analysis of the operator \(S\) is essential to the biorthogonal expansion of the symmetrizable operator \(B\).