Search results for "A-frame"
showing 4 items of 4 documents
Weak A-frames and weak A-semi-frames
2021
After reviewing the interplay between frames and lower semi-frames, we introduce the notion of lower semi-frame controlled by a densely defined operator $A$ or, for short, a weak lower $A$-semi-frame and we study its properties. In particular, we compare it with that of lower atomic systems, introduced in (GB). We discuss duality properties and we suggest several possible definitions for weak $A$-upper semi-frames. Concrete examples are presented.
Frames and weak frames for unbounded operators
2020
In 2012 G\u{a}vru\c{t}a introduced the notions of $K$-frame and of atomic system for a linear bounded operator $K$ in a Hilbert space $\mathcal{H}$, in order to decompose its range $\mathcal{R}(K)$ with a frame-like expansion. In this article we revisit these concepts for an unbounded and densely defined operator $A:\mathcal{D}(A)\to\mathcal{H}$ in two different ways. In one case we consider a non-Bessel sequence where the coefficient sequence depends continuously on $f\in\mathcal{D}(A)$ with respect to the norm of $\mathcal{H}$. In the other case we consider a Bessel sequence and the coefficient sequence depends continuously on $f\in\mathcal{D}(A)$ with respect to the graph norm of $A$.
Electrochemical reduction properties of A-frame compounds and crystal structure of Pd2(dppm)2(Me)2(Br)+ dimer
2006
Abstract Two series of A-frame complexes, [Pd2(dppm)2(R)2(μ-X)]+ (R = Me and X = Cl, Br, I, H; R = Mes and X = Br, I), were investigated by cyclic voltammetry (CV). The 2-electron reduction potentials for the first series increase from I (−1.10), Br (−1.17), Cl (−1.25) to H (−1.65 V versus SCE, in CHCl3), as well as in the second series; Br (−1.35) and I (−1.38 V versus SCE, in THF). The nature of the LUMO where the electron reduction takes place is qualitatively addressed by DFT on the corresponding model complexes [Pd2(H2PCH2PH2)2(R)2(μ-X)]+. The LUMO and (LUMO + 1) of the halide derivatives exhibit the presence of Pd d x 2 - y 2 atomic orbitals interacting in an anti-bonding fashion with…
Continuous frames for unbounded operators
2021
Few years ago G\u{a}vru\c{t}a gave the notions of $K$-frame and atomic system for a linear bounded operator $K$ in a Hilbert space $\mathcal{H}$ in order to decompose $\mathcal{R}(K)$, 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 on a Hilbert space $A$ in a continuous setting, thus extending what have been done in a previous paper in a discrete framework.