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Special classes of intertwining transformations between Hilbert spaces are introduced and investigated, whose purposes are to provide partial answers to some classical questions on the existence of nontrivial invariant subspaces for operators acting on separable Hilbert spaces. The main result ensures that if an operator is \({{\mathcal D}}\)-intertwined to a normal operator, then it has a nontrivial invariant subspace.

Positivity of a $2\times2$ operator matrix $[\begin{smallmatrix}A&B\\B^{*}&C\end{smallmatrix}]\geq0$ implies $\sqrt{\|A\|\cdot\|C\|}\geq\|B\|$ for operator norm $\|\cdot\|$ . This can be considered as an operator version of the Schwarz inequality. In this situation, for $A,C\geq0$ , there is a natural notion of geometric mean $A\sharp C$ , for which $\sqrt{\|A\|\cdot\|C\|}\geq\|A\sharp C\|$ . In this paper, we study under what conditions on $A$ , $B$ , and $C$ or on $B$ alone the norm inequality $\sqrt{\|A\|\cdot\|C\|}\geq\|B\|$ can be improved as $\|A\sharp C\|\geq\|B\|$ .

Inspired by similar definition in subdifferential theory, we define limiting sublevel set and limiting normal operator maps for quasiconvex functions. These maps satisfy important properties as semicontinuity and quasimonotonicity. Moreover, calculus rules together with necessary and sufficient optimality conditions for constrained optimization are established.

This paper gives a characterization of the asymptotic limit AT associated to a contraction T that is similar to a normal operator (Theorem 2). Extensions from contractions to power bounded operators intertwined to a contraction with a \({\mathcal{C}_{0}}\). completely nonunitary part (not necessarily a normaloid contraction) are considered as well (Theorem 1). It is also given a characterization of the asymptotic limit AT for a hyponormal contraction T, and it is shown that if a hyponormal contraction has no nontrivial invariant subspace, then one of the defect operators is not finite-rank (Corollary 1).

In this paper, we study the Fuglede–Putnam property. We give a necessary and sufficient condition for which $$(\oplus _{i=1}^{n}A_i,\oplus _{i=1}^{n}B_i)$$ satisfies the Fuglede–Putnam property. We also study the local spectral theory associated with the Fuglede–Putnam property. Finally, we define the weak Fuglede–Putnam property and we investigate several cases which satisfy the weak Fuglede–Putnam property.

In this paper, we prove the following assertions: (1) If $T$ is a $k$-quasi-$*$-$A(n)$ operator, then $T$ is polaroid. (2) If $T$ is a $k$-quasi-$*$-$A(n)$ operator, then the spectrum $\sigma$ is continuous. (3) If $T$ or $T^{*}$ is a $k$-quasi-$*$-$A(n)$ operator, then Weyl's theorem holds for $f(T)$ for every $f \in H(\sigma(T))$. (4) If $T^{*}$ is a $k$-quasi-$*$-$A(n)$ operator, then generalized $a$-Weyl's theorem holds for $f(T)$ for every $f \in H(\sigma(T))$. Finally, the finiteness of a quasi-$*$-$A(n)$ operator is also studied.

An open question, raised independently by several authors, asks if a closed amenable subalgebra of $${\mathfrak{B}(\mathfrak{H})}$$ must be similar to an C *-algebra. Recently, Choi, Farah and Ozawa have found a counter-example to this question, but their example is neither separable nor commutative, which leaves the question open for singly-generated algebras. In this paper we continue this line of investigation for special singly-generated algebras. It is shown that if an amenable operator T = N + K, where N is a normal operator, K is a compact operator and σ e (N) has only finite accumulation points, then T is similar to a normal operator; if an amenable operator T = N + K, where N is a normal operator, $${K\in\mathcal{C}_p}$$ for some p > 1 and $${\sigma(T)\cup\sigma(N)}$$ is included in a smooth Jordan curve, then T is similar to a normal operator; if an amenable operator T = N + Q, where N is a normal operator, Q is a polynomial compact operator and NQ = QN, then T is similar to a normal operator; if there exists p, 1

Given a complex Krein space H with fundamental symmetry J, the aim of this note is to characterize the set of J-normal projections Q={Q∈L(H):Q2=QandQ#Q=QQ#}. The ranges of the projections in QQ are exactly those subspaces of HH which are pseudo-regular. For a fixed pseudo-regular subspace S , there are infinitely many J-normal projections onto it, unless SS is regular. Therefore, most of the material herein is devoted to parametrizing the set of J-normal projections onto a fixed pseudo-regular subspace S., Facultad de Ciencias Exactas, Consejo Nacional de Investigaciones Científicas y Técnicas

We prove the following generalization of the Fuglede–Puntam theorem. Let N be an unbounded normal operator in the Hilbert space, and let A be an unbounded self-adjoint operator such that \(D(N)\subseteq D(A)\). Then, \( AN\subseteq N^*A \Rightarrow AN^*\subseteq NA.\)

We prove a theorem on the optimal recovery of powers of a normal operator. To illustrate the result, we prove assertion concerning the optimal recovery of the temperature of a body in the difference model of the heat equation and the optimal recovery of a solution in the difference model of a system of ordinary differential equations. Bibliography: 6 titles.

‎An asymmetric Fuglede-Putnam's Theorem for $w$-hyponormal operators and class $\mathcal{Y}$ operators is proved‎, ‎as a consequence of this result‎, ‎we obtain that the range of the generalized derivation induced by the above classes of‎ ‎operators is orthogonal to its kernel‎.

We give the definition of an abstract Mittag-Leffler kernel ℰρ ranging in a separable Hilbert space ℌ. In the simplest case, ℰ ρ (z) can be expressed via the Mittag-Leffler function ℰ ρ (z, µ). The kernel ℰ is said to be c-regular if it generates an integral transform of Fourier- Dzhrbashyan type and d-regular if its range contains an unconditional basis of ℌ. We give a complete description of d- and c-regular kernels, which permits us to answer a question posed by M. Krein. An application to the problem on the similarity of a rank one perturbation of a fractional power of a Volterra operator to a normal operator is considered.

The Aluthge transform \( \widetilde{T} \) (defined below) of an operator T on Hilbert space has been studied extensively, most often in connection with p-hyponormal operators. In [6] the present authors initiated a study of various relations between an arbitrary operator T and its associated \( \widetilde{T} \), and this study was continued in [7], in which relations between the spectral pictures of T and \( \widetilde{T} \) were obtained. This article is a continuation of [6] and [7]. Here we pursue the study of the sequence of Aluthge iterates {\( \widetilde{T} \)(n)} associated with an arbitrary operator T. In particular, we verify that in certain cases the sequence {\( \widetilde{T} \)(n)} converges to a normal operator, which partially answers Conjecture 1.11 in [6] and its modified version below (Conjecture 5.6).

We show that a pluripotential proof of the uniform estimate in the Calabi-Yau theorem works also in the Hermitian case.

In this article we explore the concept of the normal cone to the sublevel sets (or strict sublevel sets) of a function. By slightly modifying the original definition of Borde and Crouzeix, we obtain here a new (but strongly related to the already existent) notion of a normal operator. This technique turns out to be appropriate in Quasiconvex Analysis since it allows us to reveal characterizations of the various classes of quasiconvex functions in terms of the generalized quasimonotonicity of their `normal' multifunctions.

For bounded normal operators in Krein spaces we give a necessary and sufficient condition for strong stability. The same result for unitary operators was obtained by M.G.Krein [1] (see also [2]). For selfadjoint operators we refer to the papers of P.Jonas, H.Langer [3] and H.Langer [4].

We investigate the commutation between a continuous linear random operator and a continuous linear deterministic operator on a Banach space. From this we obtain probabilistic versions of theorems by Fuglede and Putnam, both of them dealing with the commutation between continuous linear operators with continuous normal operators on a Hilbert space.

Let T or T* be an algebraically quasi-paranormal operator acting on a Hilbert space. We prove: (i) generalized Weyl's theorem holds for f(T) for every f ∈ H(σ (T)); (ii) generalized a-Browder's theorem holds for f(S) for every S ≺ T and f ∈ H(σ(S)); (iii) the spectral mapping theorem holds for the B-Weyl spectrum of T. Moreover, we show that if T is an algebraically quasi-paranormal operator, then T + F satisfies generalized Weyl's theorem for every algebraic operator F which commutes with T.

A complete characterization of normal operators which are similar to irreducible operators and some related results are given.

The authors announce a newly-proved theorem of theirs. This theorem is of principal significance to numerical computation of operator equations of the first kind.

Let T be a continuous linear operator in LF-spaces. In [3], Theorem (2.6) we gave sufficient conditions for normal solvability of the adjoint T. Here we will prove that the conditions stated under B in this theorem are also necessary if N(T) is orthogonal in X.

It is shown that the effective non-hermitean one-particle Hamiltonian Heff, deduced from many-body theory, is a measurable operator only if Heff is a normal operator; if Heff is not a normal operator, quasiparticles and the frequency-dependent quasiparticle-eigenfunctions are defined by the hermitean spectral-operator. Using perturbation-theory, an estimation of the frequency-dependent parts of the core-electron eigenfunctions is given which shows the limitations in defining a core-electron Hilbert subspace.

It is known that ifA is a normal reductive linear operator on a Hilbert space and the linear system {A, B} is controllable, then there exists a vectorb in the range ofB such that {A, B} is controllable. It is shown that this result does not hold for an arbitrary normal operatorA.