Abstract
This paper deals with the possibility of transforming a weakly measurable function in a Hilbert space into a continuous frame by a metric operator, i.e., a strictly positive self-adjoint operator. A necessary condition is that the domain of the analysis operator associated with the function be dense. The study is done also with the help of the generalized frame operator associated with a weakly measurable function, which has better properties than the usual frame operator. A special attention is given to lower semi-frames: indeed, if the domain of the analysis operator is dense, then a lower semi-frame can be transformed into a Parseval frame with a (special) metric operator.
Highlights
In recent papers, one of us (RC) [21,22] has analyzed sesquilinear forms defined by sequences in Hilbert spaces and operators associated with them by means of representation theorems
A weakly measurable function φ is a continuous frame of H if there exist constants 0 < m ≤ M < ∞, such that: m f 2 ≤ | f |φx |2 dμ(x) ≤ M f 2, ∀ f ∈ H
After reviewing the conventional definitions about frames and semi-frames in Sect. 2, we introduce in Sect. 3 the generalized frame operator Tφ, whose properties are more convenient that those of the standard frame operator Sφ
Summary
One of us (RC) [21,22] has analyzed sesquilinear forms defined by sequences in Hilbert spaces and operators associated with them by means of representation theorems. He derived results about lower semi-frames and duality. A weakly measurable function φ is said to be μ-total if f |φx = 0 for a.e. x ∈ X implies that f = 0. A weakly measurable function φ is a continuous frame of H if there exist constants 0 < m ≤ M < ∞ (the frame bounds), such that:. M f 2 ≤ | f |φx |2 dμ(x) ≤ M f 2 , ∀ f ∈ H
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