Operator-valued frame generators for group-like unitary systems (Q2852262)

This paper is concerned with operator-valued frames. An operator-valued frame is an at most countable family \(V=\{V_j\}_{j \in J}\) where each element is a bounded linear map \(V_j: {\mathcal H} \to {\mathcal H}_0\) between Hilbert spaces \(\mathcal H\) and \({\mathcal H}_0\) such that there exist constants \(a,b >0\) for which the operator inequalities NEWLINE\[NEWLINE a I \leq \sum_{j \in J} V_j^* V_j \leq b I NEWLINE\]NEWLINE hold. The analysis operator \(\theta_V\) associated with \(V\) is the map from \(\mathcal H\) to \(\ell^2(J)\otimes {\mathcal H}_0\) given by NEWLINE\[NEWLINE \theta_V(x) = \sum_{j \in J} e_j \otimes V_j(x), x \in \mathcal H \, . NEWLINE\]NEWLINENEWLINENEWLINEA first problem is to determine when an operator-valued frame \(\{V_j\}\) has a Parseval dual, which means there exists an operator-valued frame \(\{W_j\}_{j \in J}\) such that \(\sum_{j \in J} W_j^* V_j = I\) and \(\sum_{j \in J} W_j^* W_j = I\). The author shows that a necessary condition for having a Parseval dual is \(I \leq \sum_{j \in J} V_j^* V_j\). If this is satisfied and if the Hilbert space \(\mathcal H\) can be embedded isometrically in the orthogonal complement of the range of \(\theta_V\), then \(V\) has indeed a Parseval dual. This is an extension of an earlier result on frames, that is, the case when the dimension of \(\mathcal H_0\) is one, by \textit{D. Han} and \textit{D. R. Larson} [Mem. Am. Math. Soc. 697, 94 p. (2000; Zbl 0971.42023)].NEWLINENEWLINENext, the paper characterizes operator-valued frames that are obtained as orbits under group-like unitary systems. A group-like unitary system \(\mathcal U\) is an at most countable set of unitary operators on \(\mathcal H\) such that the group generated by \(\mathcal U\) is contained in the set \( \mathbb T \mathcal U = \{\lambda U: \lambda \in \mathbb C, |\lambda| =1, U \in \mathcal U \} \, . \) A unitary representation \(\pi\) of \(\mathcal U\) on \(\mathcal H\) is called an operator-valued frame representation if there exists a bounded linear map \(A\) from \(\mathcal H\) to \({\mathcal H}_0\) such that \(\{A\pi(U): U \in \mathcal U\}\) is an operator-valued frame. It is shown that \(\pi\) is a frame representation if and only if the commutant of \(\pi(\mathcal U)\) consists of all operators \(\{\theta_A^* \theta_B: A, B \in B_\pi\}\) where \(B_\pi\) is the set containing each operator \(A\) mapping from \(\mathcal H\) to \({\mathcal H}_0\) for which \(\theta_A\) is bounded.NEWLINENEWLINEFinally, the paper investigates which group-like unitary systems provide an abundance of operator-valued frames with Parseval duals. To this end, the notion of frame multiplicity is introduced in the context of operator-valued frames. This is the supremum of the number \(n\) of generators \(A_1, A_2, \dots, A_n\) such that \(\{A_i \pi(U): U \in \mathcal U\}\) are mutually orthogonal. Each operator-valued frame \(\{A \pi(U): U \in {\mathcal U}\}\) generated by \(A: \mathcal H \to {\mathcal H}_0\) with a lower frame bound greater than or equal to one has a Parseval dual if and only if \(\pi\) has frame multiplicity at least equal to two.NEWLINENEWLINEThe paper concludes with some observations on generators for operator-valued Parseval frames in the case of finite dimensional \(\mathcal H\).