Selfadjoint Operators In Hilbert Spaces Research Articles

Refinements and Reverses of Tensorial and Hadamard Product Inequalities for Selfadjoint Operators in Hilbert Spaces Related to Young's Result

Let $H$ be a Hilbert space. In this paper we show among others that, if the selfadjoint operators $A$ and $B$ satisfy the condition $0$ $<$ $m\leq A,$ $B\leq M,$ for some constants $m,$ $M,$ then \begin{align*} 0& \leq \frac{m}{M^{2}}\nu \left( 1-\nu \right) \left( \frac{A^{2}\otimes 1+1\otimes B^{2}}{2}-A\otimes B\right) \\ & \leq \left( 1-\nu \right) A\otimes 1+\nu 1\otimes B-A^{1-\nu }\otimes B^{\nu } \\ & \leq \frac{M}{m^{2}}\nu \left( 1-\nu \right) \left( \frac{A^{2}\otimes 1+1\otimes B^{2}}{2}-A\otimes B\right) \end{align*} for all $\nu \in \left[ 0,1\right] .$ We also have the inequalities for Hadamard product \begin{align*} 0& \leq \frac{m}{M^{2}}\nu \left( 1-\nu \right) \left( \frac{A^{2}+B^{2}}{2}% \circ 1-A\circ B\right) \\ & \leq \left[ \left( 1-\nu \right) A+\nu B\right] \circ 1-A^{1-\nu }\circ B^{\nu } \\ & \leq \frac{M}{m^{2}}\nu \left( 1-\nu \right) \left( \frac{A^{2}+B^{2}}{2}% \circ 1-A\circ B\right) \end{align*} for all $\nu \in \left[ 0,1\right] .$

Variations in the Tensorial Trapezoid Type Inequalities for Convex Functions of Self-Adjoint Operators in Hilbert Spaces

In this paper, various tensorial inequalities of trapezoid type were obtained. Identity from classical analysis is utilized to obtain the tensorial version of the said identity which in turn allowed us to obtain tensorial inequalities in Hilbert space. The continuous functions of self-adjoint operators in Hilbert spaces have several tensorial norm inequalities discovered in this study. The convexity features of the mapping f lead to the variation in several inequalities of the trapezoid type.

A Trapezoid Type Tensorial Norm Inequality for Continuous Functions of Selfadjoint Operators in Hilbert Spaces

A Trapezoid Type Tensorial Norm Inequality for Continuous Functions of Selfadjoint Operators in Hilbert Spaces

Tensorial and Hadamard product inequalities for functions of selfadjoint operators in Hilbert spaces in terms of Kantorovich ratio

Let H be a Hilbert space. In this paper we show among others that, if f, g are continuous on the interval I with 0 <γ ≤ f(t)/g(t)≤ Γ for t ∈ I and if A and B are selfadjoint operators with Sp (A), Sp (B) ⊂ I, then [f1−ν(A) gν (A)] ⊗ [fν(B) g1−ν (B)] ≤ (1 − ν) f (A) ⊗ g (B) + ν g (A) ⊗ f (B) ≤ [(γ + Γ) 2/4γΓ]R [f1−ν(A) gν (A)] ⊗ [fν(B) g1−ν (B)]. The above inequalities also hold for the Hadamard product “ ◦ ” instead of tensorial product “ ⊗ ”.

Differentiable Ostrowski type tensorial norm inequality for continuous functions of selfadjoint operators in Hilbert spaces

In this paper several tensorial norm inequalities of Ostrowski type for continuous functions of selfadjoint operators in Hilbert spaces have been obtained. Multiple inequalities are obtained with variations due to the convexity properties of the mapping f.

Twice Differentiable Ostrowski Type Tensorial Norm Inequality for Continuous Functions of Selfadjoint Operators in Hilbert Spaces

\( \newcommand\norm[1]{\left\lVert#1\right\rVert}\newcommand\normx[1]{\left\Vert#1\right\Vert} \)In this paper several tensorial norm inequalities for continuous functions of selfadjoint operators in Hilbert spaces have been obtained. Multiple inequalities are obtained with variations due to the convexity properties of the mapping $f$$$\norm{(1\otimes B-A\otimes 1)^{-1}[\operatorname{exp}(1\otimes B)-\operatorname{exp}(A\otimes 1)]- \operatorname{exp}\left(\frac{A\otimes 1+1\otimes B}{2}\right)}$$$$\leqslant \norm{1\otimes B-A\otimes 1}^{2}\frac{\norm{f''}_{I,+\infty}}{24}.$$

Twice Differentiable Ostrowski Type Tensorial Norm Inequality for Continuous Functions of Selfadjoint Operators in Hilbert Spaces

Twice Differentiable Ostrowski Type Tensorial Norm Inequality for Continuous Functions of Selfadjoint Operators in Hilbert Spaces

Some Refinements of the Tensorial Inequalities in Hilbert Spaces

Hermite–Hadamard inequalities and their refinements have been investigated for a long period of time. In this paper, we obtained refinements of the Hermite–Hadamard inequality of tensorial type for the convex functions of self-adjoint operators in Hilbert spaces. The obtained inequalities generalize the previously obtained inequalities by Dragomir. We also provide useful Lemmas which enabled us to obtain the results. The examples of the obtained inequalities for specific convex functions have been given in the example and consequences section. Symmetry in the upper and lower bounds can be seen in the last Theorem of the paper given, as the upper and lower bounds differ by a constant.

On a nonlocal problem for the first-order differential-operator equations

In this work, we study the spaces of generalised elements identified with formal Fourier series and constructed via a non-negative self-adjoint operator in Hilbert space. The spectrum of this operator is purely discrete. For a differential-operator equation of the first order, we formulate a nonlocal multipoint by time problem if the corresponding condition is satisfied in a positive or negative space that is constructed via such operator; such problem can be treated as a generalisation of an abstract Cauchy problem for the specified differential-operator equation. The correct solvability of the aforementioned problem is proven, a fundamental solution is constructed, and its structure and properties are studied. The solution is represented as an abstract convolution of a fundamental solution with a boundary element. This boundary element is used to formulate a multipoint condition, and it is a linear continuous functional defined in the space of main elements. Furthermore, this solution satisfies multipoint condition in a negative space that is adjoint with a corresponding positive space of elements.

The best approximation of closed operators by bounded operators in Hilbert spaces

We solve the problem of the best approximation of closed operators by linear bounded operators in Hilbert spaces under assumption that the operator transforms orthogonal basis in Hilbert space into an orthogonal system. As a consequence, sharp additive Hardy-Littlewood-Pólya type inequality for multiple closed operators is established. We also demonstrate application of these results in concrete situations: for the best approximation of powers of the Laplace-Beltrami operator on classes of functions defined on closed Riemannian manifolds, for the best approximation of differentiation operators on classes of functions defined on the period and on the real line with the weight $e^{-x^2}$, and for the best approximation of functions of self-adjoint operators in Hilbert spaces.

Relatively Bounded Perturbations of J-Non-Negative Operators

We improve known perturbation results for self-adjoint operators in Hilbert spaces and prove spectral enclosures for diagonally dominant J-self-adjoint operator matrices. These are used in the proof of the central result, a perturbation theorem for J-non-negative operators. The results are applied to singular indefinite Sturm-Liouville operators with L^p-potentials. Known bounds on the non-real eigenvalues of such operators are improved.

Fully computable a posteriori error bounds for eigenfunctions

For compact self-adjoint operators in Hilbert spaces, two algorithms are proposed to provide fully computable a posteriori error estimate for eigenfunction approximation. Both algorithms apply well to the case of tight clusters and multiple eigenvalues, under the settings of target eigenvalue problems. Algorithm I is based on the Rayleigh quotient and the min-max principle that characterizes the eigenvalue problems. The formula for the error estimate provided by Algorithm I is easy to compute and applies to problems with limited information of Rayleigh quotients. Algorithm II, as an extension of the Davis–Kahan method, takes advantage of the dual formulation of differential operators along with the Prager–Synge technique and provides greatly improved accuracy of the estimate, especially for the finite element approximations of eigenfunctions. Numerical examples of eigenvalue problems of matrices and the Laplace operators over convex and non-convex domains illustrate the efficiency of the proposed algorithms.

Modelling of derivatives pricing using methods of spectral analysis

In this article expands the method of finding the approximate price for a wide class of derivative financial instruments. Using the spectral theory of self-adjoint operators in Hilbert space and the wave theory of singular and regular perturbations, the analytical formula of the approximate asset price is established. Methods for calculating the approximate price of options using the tools of spectral analysis, singular and regular wave theory in the case of fast and slow factors are developed. Combining methods from the spectral theory of singular and regular perturbations, it is possible to estimate the price of derivative financial instruments as a schedule by eigenfunctions. The approximate value of securities and their rate of return are calculated. Applying the theory of Sturm-Liouville, Fredholm’s alternative and analysis of singular and regular perturbations at different time scales have enabled us to obtain explicit formulas for the approximate value of securities and their yield on the basis of the development of their eigenfunctions and eigenvalues of self-adjoint operators using boundary value problems for singular and regular perturbations. The theorem of closeness estimates for bond prices approximation is proved. An algorithm for calculating the approximate price of derivatives and the accuracy of estimates has been developed, which allows to analyze and draw precautionary conclusions and suggestions to minimize the risks of pricing derivatives that arise in the stock market. A model for finding the value of derivatives corresponding to the dynamics of the stock market and the size of financial flows has been developed. This model allows you to find the prices of derivatives and their volatility, as well as minimize speculative changes in pricing, analyze the progress of stock market processes and take concrete steps to improve the situation to optimize financial strategies. The used methodology of European options pricing based on the study of volatility behavior and analysis of the yield of financial instruments allows to increase the accuracy of the forecast and make sound management strategic decisions by stock market participants.

Choi–Davis–Jensen’s type trace inequalities for convex functions of self-adjoint operators in Hilbert spaces

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.

Multidimensional spectral order for selfadjoint operators

The notion of the spectral order is extended to finite families of pairwise commuting bounded and unbounded selfadjoint operators in Hilbert space. It is shown that the multidimensional spectral order ≼ is preserved by transformations represented by spectral integrals of separately increasing Borel functions on Rκ. In particular, the κ-dimensional spectral order is the restriction of product of κ spectral orders for selfadjoint operators. In the context of positive κ-tuples of pairwise commuting selfadjoint operator, the relation A≼B holds if and only if Aα⩽Bα for every κ-tuples of nonnegative integer numbers α.

Regularization of linear ill-posed problems involving multiplication operators

ABSTRACT We study regularization of ill-posed equations involving multiplication operators when the multiplier function is positive almost everywhere and zero is an accumulation point of the range of this function. Such equations naturally arise from equations based on non-compact self-adjoint operators in Hilbert space, after applying unitary transformations arising out of the spectral theorem. For classical regularization theory, when noisy observations are given and the noise is deterministic and bounded, then non-compactness of the ill-posed equations is a minor issue. However, for statistical ill-posed equations with non-compact operators less is known if the data are blurred by white noise. We develop a theory for spectral regularization with emphasis on this case. In this context, we highlight several aspects, in particular, we discuss the intrinsic degree of ill-posedness in terms of rearrangements of the multiplier function. Moreover, we address the required modifications of classical regularization schemes in order to be used for non-compact statistical problems, and we also introduce the concept of the effective ill-posedness of the operator equation under white noise. This study is concluded with prototypical examples for such equations, as these are deconvolution equations and certain final value problems in evolution equations.

Topological degree for equivariant gradient perturbations of an unbounded self-adjoint operator in Hilbert space

We present a version of the equivariant gradient degree defined for equivariant gradient perturbations of an equivariant unbounded self-adjoint operator with purely discrete spectrum in Hilbert space. Two possible applications are discussed.

COMPARISONS OF HEINZ OPERATOR MEANS WITH DIFFERENT PARAMETERS

This article aims to present new comparisons of Heinz operator means with different parameters by the help of appropriate scalar comparisons and the monotonicity principle for bounded self-adjoint Hilbert space operators. In particular, for any positive operators, we establish the inequality where satisfying

Operator Popoviciu’s inequality for superquadratic and convex functions of selfadjoint operators in Hilbert spaces

AbstractIn this work, an operator version of Popoviciu’s inequality for positive operators on Hilbert spaces under positive linear maps for superquadratic functions is proved. Analogously, using the same technique, an operator version of Popoviciu’s inequality for convex functions is obtained. Some other related inequalities are also presented.

Spectral study of options based on CEV model with multidimensional volatility

This article studies the derivatives pricing using a method of spectral analysis, a theory of singular and regular perturbations. Using a risk-neutral assessment, the authors obtain the Cauchy problem, which allows to calculate the approximate price of derivative assets and their volatility based on the diffusion equation with fast and slow variables of nonlocal volatility, and they obtain a model with multidimensional stochastic volatility. Applying a spectral theory of self-adjoint operators in Hilbert space and a theory of singular and regular perturbations, an analytic formula for approximate asset prices is established, which is described by the CEV model with stochastic volatility dependent on l-fast variables and r-slowly variables, l ≥ 1, r ≥ 1, l ∈ N, r ∈ N and a local variable. Applying the Sturm-Liouville theory, Fredholm’s alternatives, as well as the analysis of singular and regular perturbations at different time scales, the authors obtained explicit formulas for derivatives price approximations. To obtain explicit formulae, it is necessary to solve 2l Poisson equations.