Lower Semi-frames, Frames, and Metric Operators

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.

Generalized frame operator, lower semiframes, and sequences of translates

“…\end{equation*}$$When $$\mathfrak {D}(C_\xi )$$ is dense, then $$T_\xi ^{-1/2}\xi$$ is a Parseval frame for $$\mathcal {H}$$ (a well‐known fact for a frame ξ, in which case, $$T_\xi ^{-1/2}\xi$$ is called the canonical tight frame of ξ). The considerations in this remark have been further investigated in [4].…”

Weak $A$-frames and weak $A$-semi-frames

“…If φ is a lower semi-frame of H, then its dual ψ is a Bessel mapping of H [8]. In addition, if D(C φ ) is dense, its dual ψ is an upper semi-frame.…”

“…(1) A reproducing kernel Hilbert space. We start from the example of a lower semi-frame in a reproducing kernel Hilbert space described in increasing generality in [6,8]. Let H K be a reproducing kernel Hilbert space of (nice) functions on a measure space (X, µ), with kernel function k .…”

“…By Proposition 2.2, χ φ is a frame. Actually, one can say more [8]. The norm f…”

“…But, one question remains: how does one obtain semi-frames? A standard construction is to start from an unbounded operator A and build a lattice of Hilbert spaces out of it, as described in [4] and in [8]. As we will see in Section 6 (1) and (2) below, this approach indeed generates a weak lower A-semi-frame.…”