Since their discovery in 1927, the Heisenberg Inequalities have become an icon of quantum mechanics [1]. Often inappropriately referred to as the Uncertainty Principle, these inequalities relating the standard deviations of the position and momentum observables to Planck’s constant are one of the cornerstones of the quantum formalism even if the physical interpretation of quantum mechanics rema...

Table of Contents Introduction—1 §1 The Dirac operator on the Hyperbolic disk A covering of the frame bundle—4 The Dirac operator—7 Covariance of the Dirac operator—10 §2 Local Expansions Eigenfunctions for infinitesimal rotations—14 Differentiating local expansions with respect to the branch points—21 Estimates at infinity—24 §3 L 2 existence results A model for the simply connected covering D...

Some necessary and sufficient conditions are given for the existence of a G-positive (G-repositive) solution to adjointable operator equations $AX=C,AXA^{left( astright) }=C$ and $AXB=C$ over Hilbert $C^{ast}$-modules, respectively. Moreover, the expressions of these general G-positive (G-repositive) solutions are also derived. Some of the findings of this paper extend some known results in the...

Abstract. In this paper, we mainly parameterize g-frames in Hilbert space in terms of operators. Firstly, under the condition that there exists a g-orthonormal basis we establish the one to one correspondence between different kinds of g-frames with certain kinds of operators. Then we parameterize g-frames without the above restriction. Finally, we consider some special g-frames, and their oper...

Abstract. The construction of frames for a Hilbert space H can be equated to the decomposition of the frame operator as a sum of positive operators having rank one. This realization provides a different approach to questions regarding frames with particular properties and motivates our results. We find a necessary and sufficient condition under which any positive finite-rank operator B can be e...

The notion of operator-valued frames (OVFs) by Kaftal et al. (Trans Am Math Soc 361(12):6349–6385, 2009) and G-frames Sun (J Anal Appl 322(1):437–452, 2006) do not include all generalizations notions for Hilbert spaces. For this purpose, we introduce the weak (weak OVFs) which covers known Theory weak-OVFs is more demanding than theory OVFs due to following: One, frame operator may factor anoth...

for all x∈H . When A = B the frame is said to be tight and if in addition, A = B = 1 it is termed a Parseval frame. When F = { fi}i=1 is a frame, we shall abuse notations and denote by F again, the n×M matrix whose ith column is fi, and where n is the dimension of H . Using this notation, the frame operator is the n×n matrix S=FF∗ where F∗ is the adjoint of F . It is a folklore to note that F i...

A generalization of the known results in fusion frames and $g$-frames theory to continuous fusion frames which defined by M. H. Faroughi and R. Ahmadi, is presented in this study. Continuous resolution of the identity (CRI) is introduced, a new family of CRI is constructed, and a number of reconstruction formulas are obtained. Also, new results are given on the duality of continuous fusion fram...