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Доведено можливість інтегрального зображення додатно визначеного ядра від двох пар змінних. Використано техніку побудови за цим ядром нового гільбертового простору, у якому формально комутують симетричні диференціальні оператори. При цьому ядро задовольняє систему диференціальних рівнянь із частинними похідними. Відомо, що ядро, задане в підобласті дійсної площини, не завжди припускає продовження на всю площину. Така можливість зумовлена проблемою існування комутувального самоспряженого розширення симетричних операторів. Застосовано результати, отримані автором, пов’язані з комутувальним самоспряженим розширенням у більш широкому гільбертовому просторі. Одержане інтегральне зображення за спектральною мірою, породженою розкладом одиниці операторів, дає змогу продовження додатно визначеного ядра на всю площину. The paper proposes proof of the possibility of an integral representation of a positive definite kernel of two pairs of variables. Using this kernel, we use the technique of constructing a new Hilbert space in which symmetric differential operators formally commute. In this case, the kernel satisfies a system of differential equations with partial derivatives. It is known that a kernel given in a subdomain of the real plane, generally speaking, does not always imply an extension to the entire plane. This possibility is related to the problem of the existence of a commuting selfadjoint extension of symmetric operators. The author applies his own results related to a commuting self-adjoint extension in a wider Hilbert space. The resulting representation in the form of an integral of elementary positive-definite kernels with respect to the spectral measure generated by the resolution of the identity of the operators allows us to extend the positive-definite kernel to the entire plane.

In this article we investigate some basic results of Self-adjoint Operator in Γ-Hilbert space. We proof some similar results on Self-adjoint Operator in this space with some specific norm. Finally we will prove that the Spectral Theorem for Compact Self-adjoint Operator in Γ -Hilbert space and the converse is true.

The main aim of this paper is to provide some range-type criteria for the essentially self-adjointness of symmetric linear relations in real or complex Hilbert spaces. The main tool is a matrix whose entries are certain linear relations.

Some bounded operators are part of this paper.Through this paper we shall obtain common properties of Some bounded operators in Г-Hilbert space. Also, introduced 2-self-adjoint operators and it’s spectrum in Г-Hilbert Space. Characterizations of these operators are also part of this literature.

Some Choi–Davis–Jensen’s type trace inequalities for convex functions are proved. Also, we generalize these inequalities for any arbitrary operator mean via operator monotone decreasing functions. In particular, we present some new order among $$\mathrm{tr}(\Phi (C)A)$$ and $$\mathrm{tr}(\Phi (C)A^{-1})$$ . New refinements of some power type trace inequalities via reverse and refinement of Young’s inequality are established. Among our results, we obtain new versions of the Holder type trace inequality for any arbitrary operator mean.

In the first part of this note we give a rather short proof of a generalization of Stenger’s lemma about the compression $$A_0$$ to $${{\mathfrak {H}}}_0$$ of a self-adjoint operator A in some Hilbert space $${{\mathfrak {H}}}={{\mathfrak {H}}}_0\oplus {{\mathfrak {H}}}_1$$ . In this situation, $$S:=A\cap A_0$$ is a symmetry in $${{\mathfrak {H}}}_0$$ with the canonical self-adjoint extension $$A_0$$ and the self-adjoint extension A with exit into $${{\mathfrak {H}}}$$ . In the second part we consider relations between the resolvents of A and $$A_0$$ like M.G. Krein’s resolvent formula, and corresponding operator models.

Let H be a separable complex Hilbert space with dim H ≥ 3, Bs(H) be the Lie algebra of all bounded self-adjoint operators on H, and let F:iBs(H)→[d,∞] with d≥ 0 be a radial unitary similarity invar...

Let A be a symmetric linear relation in the Hilbert space ℌ with unequal deficiency indices n−A < n+(A). A self-adjoint linear relation $$ \tilde{A}\supset A $$ in some Hilbert space $$ \tilde{\mathrm{\mathfrak{H}}}\supset \mathrm{\mathfrak{H}} $$ is called an (exit space) extension of A. We study the compressions $$ C\left(\tilde{A}\right)={P}_{\mathrm{\mathfrak{H}}}\tilde{A}\upharpoonright \mathrm{\mathfrak{H}} $$ of extensions $$ \tilde{A}=\tilde{A^{\ast }}. $$ Our main result is a description of compressions $$ C\left(\tilde{A}\right) $$ by means of abstract boundary conditions, which are given in terms of a limit value of the Nevanlinna parameter τ(λ) from the Krein formula for generalized resolvents. We describe also all extensions $$ \tilde{A}=\tilde{A^{\ast }}. $$ of A with the maximal symmetric compression $$ C\left(\tilde{A}\right) $$ and all extensions $$ \tilde{A}=\tilde{A^{\ast }}. $$ of the second kind in the sense of M.A. Naimark. These results generalize the recent results by A. Dijksma, H. Langer and the author obtained for symmetric operators A with equal deficiency indices n+(A) = n−(A).

Define $ A $ a unbounded self-adjoint operator on Hilbert space $ X $. Let $ \{ A_n \} $ be its resolvent approximation sequence with closed range $ \mathcal{R}(A_n) (n \in \mathrm{N}) $, that is, $ A_n (n \in \mathrm{N}) $ are all self-adjoint on Hilbert space $ X $ and \begin{equation*} \hbox{ \raise-2mm\hbox{$\textstyle s-\lim \atop \scriptstyle {n \to \infty}$}} R_��(A_n) = R_��(A)\quad (��\in \mathrm{C} \setminus \mathrm{R}), \ \textrm{where} \ R_ ��(A) := (��I-A)^{-1}. \end{equation*} The Moore-Penrose inverse $ A^\dagger_n \in \mathcal{B}(X) $ is a natural approximation to the Moore-Penrose inverse $ A^\dagger $. This paper shows that: $ A^\dagger $ is continuous and strongly converged by $ \{ A^\dagger_n \} $ if and only if $ \sup\limits_n \Vert A^\dagger_n \Vert < +\infty $. On the other hand, this result tells that arbitrary bounded computational scheme $ \{ A^\dagger_n \} $ induced by resolvent approximation $ \{ A_n \} $ is naturally instable (that is, $ \sup_n \Vert A^\dagger_n \Vert = \infty $) for any self-adjoint operator equation with non-closed range, for example, free Schr��dinger operator, Schr��dinger operator with Coulumb potential and Schr��dinger operator in model of many particles. This implies the infeasibility to globally and approximately solve non-closed range self-ajoint operator equation by resolvent approximation., Ninth Version

Let $${\mathcal {B}}_{s}({\mathcal {H}})$$ be the real linear space of all self-adjoint operators on a complex Hilbert space $${\mathcal {H}}$$ with $$\dim {\mathcal {H}}\ge 3$$. It is proved that a continuous surjective map $$\varphi $$ on $${\mathcal {B}}_{s}({\mathcal {H}})$$ preserves nonzero projections of Jordan products of two operators in both directions if and only if there exist a unitary or an anti-unitary operator U on $${\mathcal {H}}$$ and a constant $$\lambda $$ with $$\lambda ^2=1$$ such that $$\varphi (A)=\lambda {U}^*AU$$ for all $$A\in {\mathcal {B}}_{s}({\mathcal {H}})$$.

The paper interprets the concept “operator in the separable complex Hilbert space” (particalry, “Hermitian operator” as “quantity” is defined in the “classical” quantum mechanics) by that of “quantum information”. As far as wave function is the characteristic function of the probability (density) distribution for all possible values of a certain quantity to be measured, the definition of quantity in quantum mechanics means any unitary change of the probability (density) distribution. It can be represented as a particular case of “unitary” qubits. The converse interpretation of any qubits as referring to a certain physical quantity implies its generalization to non-Hermitian operators, thus neither unitary, nor conserving energy. Their physical sense, speaking loosely, consists in exchanging temporal moments therefore being implemented out of the space-time “screen”. “Dark matter” and “dark energy” can be explained by the same generalization of “quantity” to non-Hermitian operators only secondarily projected on the pseudo-Riemannian space-time “screen” of general relativity according to Einstein's “Mach’s principle” and his field equation.

In applications, many states given for a system can be expressed by orthonormal elements, called “state elements,” taken in a separable Hilbert space (called “state space”). The exact nature of the Hilbert space depends on the system, for example, the state space for position and momentum states is the space of square-integrable functions. The symmetries of a quantum system can be represented by a class of unitary operators that act in the Hilbert space. The operators called ladder operators have the effect of lowering or raising the energy of the state. In this paper, we study the spectral properties of a self-adjoint, fourth-order differential operator with a bounded operator coefficient and establish a second regularized trace formula for this operator.

Let H be a Hilbert space. Suppose A is a positive self-adjoint operator on H and φ ∈ H is a cyclic unit vector. For each λ ∈ R , we can define the rank one perturbation of A by A λ = A + λ 〈 φ , ⋅ 〉 φ . To each A λ we can consider the spectral measure of φ, which we denote by μ λ . This generates a family of measures, { μ λ } , and we analyze the packing dimension of this family. Past results have determined that the Hausdorff dimension of this family can be determined if the limit inferior of a ratio involving μ is constant on a Lebesgue typical set. This ratio is sometimes called the pointwise dimension of μ and is related to the upper derivative of μ. Work has been done to make a similar argument for the packing dimension, but with little success. Using the theory of rank one perturbations and Borel transforms, we introduce the concept of Lebesgue exact dimension for μ, which allows us to determine the packing dimension of spectral measures of almost every rank one perturbation μ λ . If the Lebesgue exact dimension for μ is 1 α 2 then the packing dimension of Lebesgue almost every μ λ is 2 − α . As a corollary, we find that this limit condition implies a stronger result: the Hausdorff and packing dimensions are equal for almost every μ λ .

In this paper, it is obtained some new inequalities for operator m-convex functions, which they are continuous functions of self adjoint operators, in Hilbert space. Later, it is established some applications for synchronous and asynchronous functions with respect to operator m-convex functions.

In this paper we study spectral properties of a linear non-self-adjoint operator with an internalsymmetry of the form:;∗ܮ ൌ ܮ ,∗ܮ ൌ ܮwhere ∗ ൌ , ∗ ൌ are orthogonal projections, ܮ ∗is an operator, adjoint to the operator ܮ in the Hilbert space ܪ .It is shown that a spectrum of such operator is real. In the case of a discrete operator, with a complete system ofeigenvectors and associated vectors, the projections of eigenvalues and associated vectors of the operator L and itsadjoint operator form an orthonormal basis. A class of Sturm-Liouville operators with such symmetry is found,moreover, it is found that the characteristic function of such an operator factorizes. An illustrative example isprovided.

In defining the finiteness or infiniteness conditions of discrete spectrum of the Schrodinger operators, a fundamental understanding on $$n(1, F(\cdot ))$$ is crucial, where n(1, F) is the number of eigenvalues of the Fredholm operator F to the right of 1. Driven by this idea, this paper provided the invertibility condition for some class of operators. A sufficient condition for finiteness of the discrete spectrum involving the self-adjoint operator acting on Hilbert space was achieved. A relation was established between the eigenvalue 1 of the self-adjoint Fredholm operator valued function $$F(\cdot )$$ defined in the interval of (a, b) and discontinuous points of the function $$n(1, F(\cdot ))$$ . Besides, the obtained relation allowed us to define the finiteness of the numbers $$z\in (a,b)$$ for which 1 is an eigenvalue of F(z) even if $$F(\cdot )$$ is not defined at a and b. Results were validated through some examples.

[EN] The aim of this work is to study the structure of bounded finite potent endomorphisms on Hilbert spaces. In particular, for these operators, an answer to the Invariant Subspace Problem is given and the main properties of its adjoint operator are offered. Moreover, for every bounded finite potent endomorphism we show that Tate’s trace coincides with the Leray trace and with the trace defined by R. Elliott for Riesz Trace Class operators, Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature., Publicación en abierto financiada por el Consorcio de Bibliotecas Universitarias de Castilla y León (BUCLE), con cargo al Programa Operativo 2014ES16RFOP009 FEDER 2014-2020 DE CASTILLA Y LEÓN, Actuación:20007-CL - Apoyo Consorcio BUCLE

In this chapter, we recall some Keldysh results that are not only specific for Hilbert spaces, but they have been formulated for operators in Banach spaces. We also discuss some theorems on denseness of the generalized eigenvectors of a compact operator or an operator with compact resolvent and different conditions assuring the completeness of the system of root subspaces. In the selection of the material, we present as fully as the various existing results.

Non-self-adjoint operators have many applications, including quantum and heat equations. On the other hand, the study of these types of operators is more difficult than that of self-adjoint operators. In this paper, our aim is to study the resolvent and the spectral properties of a class of non-self-adjoint differential operators. So we consider a special non-self-adjoint elliptic differential operator (Au)(x) acting on Hilbert space and first investigate the spectral properties of space H 1 = L 2 Ω 1 . Then, as the application of this new result, the resolvent of the considered operator in ℓ -dimensional space Hilbert H ℓ = L 2 Ω ℓ is obtained utilizing some analytic techniques and diagonalizable way.

Observables in quantum mechanics are represented by self-adjoint operators on Hilbert space. Such ubiquitous, well-known, and very foundational fact, however, is traditionally subtle to be explained in typical first classes in quantum mechanics, as well as to senior physicists who have grown up with the lesson that self-adjointness is "just technical". The usual difficulties are to clarify the connection between the demand for certain physical features in the theory and the corresponding mathematical requirement of self-adjointness, and to distinguish between self-adjoint and hermitian operator not just at the level of the mathematical definition but most importantly from the perspective that mere hermiticity, without self-adjointness, does not ensure the desired physical requirements and leaves the theory inconsistent. In this work we organise an amount of standard facts on the physical role of self-adjointness into a coherent pedagogical path aimed at making quantum observables emerge as necessarily self-adjoint, and not merely hermitian operators. Next to the central core of our line of reasoning -- the necessity of a non-trivial declaration of a domain to associate with the formal action of an observable, and the emergence of self-adjointness as a consequence of fundamental physical requirements -- we include some complementary materials consisting of a few instructive mathematical proofs and a short retrospective, ranging from the past decades to the current research agenda, on the self-adjointness problem for quantum Hamiltonians of relevance in applications., 39 pages

If $T$ is a (densely defined) self-adjoint operator acting on a complex Hilbert space $\mathcal{H}$ and $I$ stands for the identity operator, we introduce the delta function operator $\lambda \mapsto \delta \left(\lambda I-T\right) $ at $T$. When $T$ is a bounded operator, then $\delta \left(\lambda I-T\right) $ is an operator-valued distribution. If $T$ is unbounded, $\delta \left(\lambda I-T\right) $ is a more general object that still retains some properties of distributions. We derive various operative formulas involving $\delta \left(\lambda I-T\right) $ and give several applications of its usage.

The paper describes approximations properties of monotonically increasing sequences of invariant subspaces of a self-adjoint operator, as well as their symmetric generalizations in a complex Hilbert space, generated by its positive powers. It is established that the operator keeps its spectrum over the dense union of these subspaces, equipped with quasi-norms, and that it is contractive. The main result is an inequality that provides an accurate estimate of errors for the best approximations in Hilbert spaces by these invariant subspaces.