Continuous frames for unbounded operators

Abstract

Few years ago Găvruţa gave the notions of K-frame and atomic system for a linear bounded operator K in a Hilbert space $$\mathcal {H}$$ H in order to decompose $$\mathcal {R}(K)$$ R ( K ) , the range of K, with a frame-like expansion. These notions are here generalized to the case of a densely defined and possibly unbounded operator A on a Hilbert space in a continuous setting, thus extending what have been done in a previous paper in a discrete framework.

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

“…The following concept was introduced and studied in [10]. Definition 3.2.…”

“…Now, we introduce a structure that generalizes both concepts of lower semi-frame and weak A-frame. We follow mostly the terminology of [10] and keep the term "weak" because the notion leads to a weak decomposition of the range of the operator A (see Theorem 4.2). We begin with giving the following definitions.…”

“…This is a generalization of the notion of weak G-dual in [10]. , x ∈ X n , ∀n ∈ N.…”

“…Generalizing the notion of continuous weak atomic system for A [10], we consider the following: Definition 4.6. Let A be a densely defined operator on H. A lower atomic system for A is a function φ : x ∈ X → φ x ∈ H such that (i) for all u ∈ D(A * ), the map x → u|φ x is a measurable function on X;…”

“…In general, the domain of C ψ is not dense, hence C * ψ is not well-defined. An example of function whose analysis operator is densely defined can be found in [10,Example 2.8], where D(C ψ ) coincides with the domain of a densely defined sesquilinear form associated to ψ. Moreover, a sufficient condition for D(C ψ ) to be dense in H is that ψ x ∈ D(C ψ ) for every x ∈ X, see [3,Lemma 2.3].…”