$K$-frames as a generalization of frames were introduced by L. Gu{a}vruc{t}a to study atomic systems on Hilbert spaces which allows, in a stable way, to reconstruct elements from the range of the bounded linear operator $K$ in a Hilbert space. Recently some generalizations of this concept are introduced and some of its difference with ordinary frames are studied. In this paper, we give a new ge...

In this paper, we establish several characterizations of the A-parallelism bounded linear operators with respect to seminorm induced by a positive operator A acting on complex Hilbert space. Among other things, investigate relationship between A-seminorm-parallelism and A-Birkhoff–James orthogonality A-bounded operators. particular, characterize which satisfy A-Daugavet equation. addition, rela...

It is known that positive operators Ω on a Hilbert space admit a factorization of the form Ω = W∗W, where W is an outer operator whose matrix representation is upper. As upper Hilbert space operators have an interpretation of transfer operators of linear time-varying systems in discrete time, this proves the existence of a spectral factorization for time-varying systems. In this paper, the abov...

In this paper we consider a class of distributed parameter systems (partial differential equations) determined by strongly nonlinear operator valued measures in the setting of the Gelfand triple V ↪→ H ↪→ V ∗ with continuous and dense embeddings where H is a separable Hilbert space and V is a reflexive Banach space with dual V ∗. The system is given by dx+A(dt, x) = f(t, x)γ(dt) +B(t)u(dt), x(0...

Let X,Y be normed spaces. The set of bounded linear operators is noted as L(X,Y ). Let now D = D(A) ⊂ X be a linear subspace, and A : D −→ Y a linear (not necessarily bounded!) operator. Notation: (A,D(A)) : X −→ Y Definition: G(A) := {(x,Ax) |x ∈ D} is called the graph of A. Obviously, G(A) is a linear subspace of X × Y . The linear operator A is called closed if G(A) is closed in X × Y . The ...

Recently, Tautenhahn and Hämarik (1999) have considered a monotone rule as a parameter choice strategy for choosing the regularization parameter while considering approximate solution of an ill-posed operator equation Tx =y , where T is a bounded linear operator between Hilbert spaces. Motivated by this, we propose a new discrepancy principle for the simplified regularization, in the setting of...

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