Non-negative Integer Power of a Hyponormal m-isometry is Reflexive

Sarason did pioneer work on reflexive operator and reflexivity of normal operators, however, he did not used the word reflexive but his results are equivalent to say that every normal operator is reflexive. The word reflexive was suggested by HALMOS and first appeared in H. Rajdavi and P. Rosenthals book `Invariant Subspaces’ in 1973. This line of research was continued by Deddens who showed that every isometry in B(H) is reflexive. R. Wogen has proved that `every quasi-normal operator is reflexive’. These results of Deddens, Sarason, Wogen are particular cases of theorem of Olin and Thomson which says that all sub-normal operators are reflexive. In other direction, Deddens and Fillmore characterized these operators acting on a finite dimensional space are reflexive. J. B. Conway and Dudziak generalized the result of reflexivity of normal, quasi-normal, sub-normal operators by proving the reflexivity of Vonneumann operators. In this paper we shall discuss the condition under which m-isometries operators turned to be reflexive.

Maximal Regularity for Non-autonomous Evolutionary Equations

We discuss the issue of maximal regularity for evolutionary equations with non-autonomous coefficients. Here evolutionary equations are abstract partial-differential algebraic equations considered in Hilbert spaces. The catch is to consider time-dependent partial differential equations in an exponentially weighted Hilbert space. In passing, one establishes the time derivative as a continuously invertible, normal operator admitting a functional calculus with the Fourier–Laplace transformation providing the spectral representation. Here, the main result is then a regularity result for well-posed evolutionary equations solely based on an assumed parabolic-type structure of the equation and estimates of the commutator of the coefficients with the square root of the time derivative. We thus simultaneously generalise available results in the literature for non-smooth domains. Examples for equations in divergence form, integro-differential equations, perturbations with non-autonomous and rough coefficients as well as non-autonomous equations of eddy current type are considered.

True Amplitude depth migration using Curvelets

We propose a true amplitude solution to the seismic imaging problem. We derive a diagonal scaling approach for the normal operator approximation in the curvelet domain. This is based on the theorem which states that curvelets remain approximately invariant under the action of the normal operator. We use curvelets as essential tools for both approximation and inversion. We also exploit the theorem which states that curvelet-domain approximation should be smooth in phase space by enforcing smoothness of curvelet coefficients in angle and space domain.We analyze our method using a reverse time migration-demigration code, simulating the acoustic wave equation on different synthetic models. Our method produces a good resolution with reflecting dips and reproduces true amplitude reflectors and compensates for incomplete illumination in seismic images.

On (P*-N) quasi normal operators Of order "n" In Hilbert space

Through this paper, we submitted some types of quasi normal operator is called be (k*-N)- quasi normal operator of order n defined on a Hilbert space H, this concept is generalized of some kinds of quasi normal operator appear recently form most researchers in the field of functional analysis, with some properties and characterization of this operator as well as, some basic operation such as addition and multiplication of these operators had been given, finally the relationships of this operator proved with some examples to illustrate conversely and introduce the sufficient conditions to satisfied this case with other types had been studied.

Some problems of approximation theory for powers of normal operators in Hilbert space

The best approximation of unbounded operator $A^k$ in class with $\| A^r x \| \leqslant 1$ and the best approximation of class with $\|A^k x \| \leqslant 1$ by class with $\| A^r x \| \leqslant N$, $N > 0$ for powers $k < r$ of normal operator $A$ in the Hilbert space $H$ are found.

Inequality of Taikov type for powers of normal operators in Hilbert space

The Taikov inequality, which estimates $L_{\infty}$-norm of intermediate derivative by $L_2$-norms of a function and its higher derivative, is extended on arbitrary powers of normal operator acting in Hilbert space.

Noncommutative Functional Calculus and Its Applications on Invariant Subspace and Chaos

Let T:H→H be a bounded linear operator on a separable Hilbert space H. In this paper, we construct an isomorphism Fxx*:L2(σ(|T−a|),μ|T−a|,ξ)→L2(σ(|(T−a)*|),μ|(T−a)*|,Fxx*Hξ) such that (Fxx*)2=identity and Fxx*H is a unitary operator on H associated with Fxx*. With this construction, we obtain a noncommutative functional calculus for the operator T and Fxx*=identity is the special case for normal operators, such that S=R|(S−a)|,ξ(Mzϕ(z)+a)R|S−a|,ξ−1 is the noncommutative functional calculus of a normal operator S, where a∈ρ(T), R|T−a|,ξ:L2(σ(|T−a|),μ|T−a|,ξ)→H is an isomorphism and Mzϕ(z)+a is a multiplication operator on L2(σ(|S−a|),μ|S−a|,ξ). Moreover, by Fxx* we give a sufficient condition to the invariant subspace problem and we present the Lebesgue class BLeb(H)⊂B(H) such that T is Li-Yorke chaotic if and only if T*−1 is for a Lebesgue operator T.

Posed Inverse Problem Rectification Using Novel Deep Convolutional Neural Network

The proposed paper addresses the inverse problems using a novel deep convolutional neural network (CNN). Over the years, regularized iterative algorithms have been observed to be the standard approach to address this issue. Though these methodologies give an excellent output, they still impose challenges such as difficulty of hyper parameter selection, increasing computational cost for adjoint operators and forward operators. It has been observed that when the normal operator of the forward model is seen to be a convolution, unrolled iterative methods take up the CNN form. In view of this observation we have proposed a methodology which uses CNN after direct inversion to find the solution for convolutional inverse problem. In the first step the physical model of the system is analyzed using direct inversion. However, this leads to artifacts which are then removed using a combination of residual learning and multi-resolution decomposition in CNN. The results show that the performance of the proposed work outperforms other algorithm and requires a maximum of 1 second to reconstruct an image of high definition.

Intuitionistic Fuzzy Normal Operator on IFH - Space

In this article, we define Intuitionistic Fuzzy Normal Operator operating on an IFH-Space. An operator is an intuitionistic fuzzy normal operator if i.e. commutes with its intuitionistic fuzzy adjoint.