Lower Semi-frames, Frames, and Metric Operators

The purpose of this paper is to propose a definition of continuous frames of rank n for Krein spaces and to study their basic properties. Similarly to the Hilbert space case, continuous frames are characterized by the analysis, the pre-frame and the frame operator, where the latter gives rise to a frame decomposition theorem. The paper includes a discussion of similar, dual and Parseval frames and of reproducing kernels. In addition, the importance of the fundamental symmetry in the formula for the frame operator in a Krein space is clarified. As prime examples, it is shown how to transfer continuous frames for Hilbert spaces to Krein spaces arising from a possibly non-regular Gram operator.

Equivalence frames and their duals in Hilbert spaces Azadeh Alijani, Zohreh Rahbani Abstract Theory of frames has very applications to various areas of science and engineering. The paper presents all duals of a constructed frame that is obtained by an operator and a primary frame. By two frames and their frame operators, a frame is given that has an identify frame operator. Moreover, a relation between ordinary duals and operator duals is given. Finally, an equation is introduced for pre-frame operator of a given frame and pre-frame operator of its operator dual. Full Text: PDF DOI: 10.15640/arms.v3n2a7

Continuous framings for Banach spaces

As a starting point we assume to have a continuous frame in a Hilbert space with respect to a measure space. This frame inherits a unitary structure from a unitary representation of a locally compact abelian group in the Hilbert space. In this setting we state a continuous sampling result for the range space of the associated analysis frame operator. The data sampling are functions also defined by using the underlying unitary structure. The result is illustrated by using continuous frames in Paley–Wiener and shift-invariant spaces generated by translates of fixed functions. A sampling strategy working only for discrete abelian groups is also discussed.

In this present paper we introduce weaving Hilbert space frames in the continuous case, we propose new approaches for manufacturing pairs of woven continuous frames, and we obtain new properties in continuous weaving frame theory related to dual frames. Also, we provide some approaches for constructing continuous weaving frames by using small perturbations. These methods not only enhance the understanding of frame theory but also open avenues for practical applications in signal processing and data representation. Future research may further explore the implications of these findings in more complex systems and their potential interdisciplinary benefits.

In Bagarello and Kużel (Acta Appl Math 171:4, 2021) Parseval frames were used to define bounded Hamiltonians, both in finite and in infinite dimesional Hilbert spaces. Here we continue this analysis, with a particular focus on the discrete spectrum of Hamiltonian operators constructed as a weighted infinite sum of rank one operators defined by some Parseval frame living in an infinite dimensional Hilbert space. The main difference with Bagarello and Kużel (Acta Appl Math 171:4, 2021) is that, here, the operators we consider are mostly unbounded. This is an useful upgrade with respect to our previous results, since physically meaningful Hamiltonians are indeed often unbounded. However, due to the fact that frames (in general) are not bases, the definition of an Hamiltonian is not so easy, and part of our results goes in this direction. Also, we analyze the eigenvalues of the Hamiltonians, and we discuss some physical applications of our framework.

In this paper we introduce approximate duality of g-frames in Hilbert $C^\\ast$-modules and we show that approximate duals of g-frames in Hilbert $C^\\ast$-modules share many useful properties with those in Hilbert spaces. Moreover, we obtain some new results for approximate duality of frames and g-frames in Hilbert spaces; in particular, we consider approximate duals of $\\varepsilon$-nearly Parseval and $\\varepsilon$-close frames.

By virtue of Furuta inequality, we show some characterizations of the spectral order of positive operators on a Hilbert space from the viewpoint of the Kan- torovich type inequality and the Riccati equation. Let A and B be positive operators on a Hilbert space. Then the spectral order AB holds if and only if there exist a unique positive contraction Tp,u such that Tp,uA p+u 2 Tp,u = A up 4 B p A up 4 for all p ≥ 0 and u ≥ 0. This form interpolates the chaotic order, δ-order and the usual order continuously. 1 Introduction. An operator means a bounded linear operator on a Hilbert space H. An operator A is said to be positive ( denoted by A ≥ 0) if (Ax, x) ≥ 0 for all x ∈ H.I n particular, we denote by A> 0i fA is positive and invertible. The usual order A ≥ B for selfadjoint operators A and B is defined by A − B ≥ 0. Let A and B be positive invertible operators. We introduce a function order A ≥f B for a real valued continuous function f on (0, ∞) defined by A ≥f B ⇐⇒ f(A) ≥ f(B).

A $J$-frame is a frame $\mathcal{F}$ for a Krein space $(\mathcal{H}, [\, , \,])$ which is compatible with the indefinite inner product $[\, , \, ]$ in the sense that it induces an indefinite reconstruction formula that resembles those produced by orthonormal bases in $\mathcal{H}$. With every $J$-frame the so-called $J$-frame operator is associated, which is a self-adjoint operator in the Krein space $\mathcal{H}$. The $J$-frame operator plays an essential role in the indefinite reconstruction formula. In this paper we characterize the class of $J$-frame operators in a Krein space by a $2\times 2$ block operator representation. The $J$-frame bounds of $\mathcal{F}$ are then recovered as the suprema and infima of the numerical ranges of some uniformly positive operators which are build from the entries of the $2\times 2$ block representation. Moreover, this $2\times 2$ block representation is utilized to obtain enclosures for the spectrum of $J$-frame operators, which finally leads to the construction of a square root. This square root allows a complete description of all $J$-frames associated with a given $J$-frame operator.

An analog of the Whittaker-Shannon-Kotel′nikov sampling theorem is derived for functions with values in a separable Hilbert space. The proof uses the concept of frames and frame operators in a Hilbert space. One of the consequences of this theorem is that it allows us to derive sampling theorems associated with boundary-value problems and some homogeneous integral equations, which in turn gives us a generalization of another sampling theorem by Kramer.

Weakly closed, commutative, symmetric rings of operators on a Hilbert space are examined and their structure is ex ploited to characterize normal operators in terms of the rings they generate. In the first chapter, notation is established, and the less common theorems from Hilbert space theory not to be used in later sections are briefly reviewed. In Chapter II, the structure of weakly closed, commu tative, symmetric rings with a cyclic vector is specified. This theorem is then used in conjunction with a strucure called a canonical decomposition system to establish that every weakly closed, commutative, symmetric ring or opera tors acting on a separable space and containing the iden tity is spatially isomorphic to a "divect sum" of L rings of a certain type. In Chapter III, two sets of necessary and sufficient conditions are given for two normal operators on a separ able Hilbert space to be unitarily equivalent. The concept of a spectral class of a normal operator is introduced -a natural generalization of the concept of an eigen value. Each spectral class is assigned a multiplicity, and it is shown that two normal operators are unitarily equivalent iff they have the same spectral classes with the same respective multiplicities. A multiplicity is iv also defined for weakly continuous positive functionals o n the symmetric ring generated by a normal operator in the weak topology, and it is shown that this multiplicity function determines the operator. This multiplicity theory is compared w i t h a previous theory. v CHAPTER I INTRODUCTION We shall be concerned in this paper with rings of operators on a Hilbert space, in particular, weakly closed, commutative, symmetric rings. All Hilbert spaces to be considered will be complex Hilbert spaces. If H is a Hilbert space and x,y e H, then (x(y) will denote the inner product of x and y, and 1 Jxi| will denote the norm of x. We shall consider only bounded linear operators in this paper, and therefore we shall use the word "operator" solely to mean a bounded linear operator. The set of all operators on the Hilbert space H will be denoted by B(H). If A s B(H), then A* will denote the adjoint of A, We employ the usual concepts of isometric, unitary, hermitian, normal, and positive definite operators. Knowledge of the weak, strong, strongest, and norm topologies for B(H) is assumed. The norm topology is the strongest of the four; hence, any set of operators closed in one of the topologies is closed in the norm topology. By a ring of operators we shall mean a set of opera tors forming a ring in the usual algebraic sense which is also a vector space over the complex numbers. A ring R of operators is symmetric if A e R whenever A e R. from the theory of Banach algebras, we know that any norm closed, 1 we define an operator A a e BCLg) by A ftf = af* Note that two Ly,, functions which differ only on a set of measure zero give rise to the same operator. The ring of operators thus obtained is a maximal, commutative, symmetric ring. If E is a weakly closed, commutative, symmetric ring with maximal ideal space M, then the fact that E is closed under sups gives rise to some strong topological properties of M. In this case, M is totally disconnected, and the closure of every open set is open. Thus, M contains many sets which are both open and closed, and such sets we call clopen. It is easy to see that projections in E correspond v i a the Gelfand transform to the characteristic functions of clopen sets. If U is a clopen subset of M, then we will let pu denote the corresponding projection in E. pu (m) -JtjjCm) = the characteristic function of U. Throughout the remainder of this discussion, E will denote a weakly closed commutative, symmetric subring of B(H) with identity, and M will denote the maximal ideal space of E. Corresponding to each vector 5 e H, there is a measure u on M defined by The topological properties of M carry strong implications for the resulting measure space. If S = M is measurable, then there exists a clopen set U such that m (S\UUU\S) * 0. If ? is cyclic for E ' , then u is supported on all of M, in ( A $ 0 = J A (m)du(m)

On lifting of operators to Hilbert spaces induced by positive selfadjoint operators

A frame in a Hilbert space H allows every element in H to be written as a linear combination of the frame elements, with coefficients called frame coefficients. Calculations of those coefficients and many other situations where frames occur, require knowledge of the inverse frame operator. But usually it is hard to invert the frame operator if the underlying Hilbert space is infinite dimensional. We introduce a method for approximation of the inverse frame operator using finite subsets of the frame. In particular this allows to approximate the frame coefficients (even in l 2—sense) using finite-dimensional linear algebra. We show that the general method simplifies when the frame contains a Riesz basis.

Heat transfer problems with axial conduction in a circular tube for an arbitrary velocity field and a semi-infinite axial domain are investigated by the linear operator method. Since the domain of the linear operator is dense in a Hilbert space, an approximation method in a Hilbert space is applied and an algebraic characteristic equation for calculating the eigenvalues is obtained. Recursive formulas for computing the expansion coefficients with alternative forms are also presented. The computational procedures are straightforward and simple. The computational results show that the asymptotic Nusselt number is insensitive to the Peclet number provided Pe is larger than about 10.

Heat transfer problems with axial conduction in a circular tube for an arbitrary velocity field and a semi-infinite axial domain are investigated by the linear operator method. Since the domain of the linear operator is dense in a Hilbert space, an approximation method in a Hilbert space is applied and an algebraic characteristic equation for calculating the eigenvalues is obtained. Recursive formulas for computing the expansion coefficients with alternative forms are also presented. The computational procedures are straightforward and simple. The computational results show that the asymptotic Nusselt number is insensitive to the Peclet number provided Pe is larger than about 10.