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In this paper, we prove that for a right linear bounded operator on a quaternionic Hilbert space, the norm and the numerical radius are equal if and only if the norm and the spectral radius are equ...

In this paper, we give new results for the p-numerical radii wp⋅ of Hilbert space operators. It is shown, among other inequalities, that if A is a Hilbert space operator, which belongs to the Schat...

In this note, we establish the cartesian decomposition of C-normal operator on a complex separable Hilbert space. Using this we give two characterizations of C-normal operators. These characterizat...

Let T be a normal bounded operator on a Hilbert space and let ω(T) denote the numerical radius of T. In this paper, we give new inequalities numerical radius of normal operatos on a Hilbert space, ...

We have developed 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 that states that curvelets remain approximately invariant under the action of the normal operator. We use curvelets as essential tools for approximation and inversion. We also exploit the theorem that states that the curvelet-domain approximation should be smooth in phase space by enforcing the smoothness of curvelet coefficients in the angle and space domains. 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, reproduces the true amplitude reflectors, and compensates for incomplete illumination in seismic images.

Let B(H) denote the algebra of all bounded linear operators acting on a complex Hilbert space H. For A,B∈B(H), define the bimultiplication operator M2,A,B on the class of Hilbert–Schmidt operators ...

The class of polynomially normal operators is a wider class than the class of all normal operators. Inspired by some interesting well known facts about normal operators and by some recent work, we ...

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.

In this paper, we give a characterization of a complex symmetric Toeplitz operator Tφ on the weighted Bergman space Aα2(D). We first give properties of complex symmetric Toeplitz operators Tφ on Aα...

A generalized Nash equilibrium problem corresponds to a noncooperative interaction between a finite set of players in which the cost function and the feasible set of each player depend on the decis...

The paper deals with the problem of operator’s state estimating [1]. For this purpose various approaches based on using deep convolutional neural networks [2, 3] are proposed. The approach using automatic emotion recognition methods and analysis of the blink characteristics are considered in the most detail. Extensive research is being done into the emotional state and attention concentrating. On the trained neural network there was conducted an experiment to estimate the operator’s state. During the experiment 11 video records of the operator’s face received during operator performing the flight task on the flight simulator [4] were processed. In the normal operator’s state during the piloting there were 2 emotions observed – sadness and a neutral state. It should be noted that the operator’s yawn and closed eyes were classified as surprise or fear. It was significant to note that the emotional background of the operators is different. When choosing parameters for determining fatigue, an individual approach is required. Experiments also are being conducted to identify the correlation between blink parameters and piloting accuracy characteristics. It was also shown that the frequency of the operator blinking decreases due to the perception of new information and concentration.

If the integrals of a one-form over all lines meeting a small open set vanish and the form is closed in this set, then the one-form is exact in the whole Euclidean space. We obtain a unique continuation result for the normal operator of the X-ray transform of one-forms, and this leads to one of our two proofs of the partial data result. Our proofs apply to compactly supported covector-valued distributions., Comment: 15 pages, 1 figure. Final version

In this paper, we introduce a new type of Drazin invertible operator on Hilbert spaces, which is called D-operator. Then, some properties of the class of D-operators are studied. We prove that the D-operator preserves the scalar product, the unitary equivalent property, the product and sum of two D-operators are not D-operator in general but the direct product and tenser product is also D-operator.

In this paper, we completely characterize invertible, unitary and isometric operators . Then we obtain the numerical range of compact normal operator . Next, we find a conjugation for the normal op...

Let H \mathcal {H} be a complex, separable Hilbert space and let B ( H ) \mathcal {B}(\mathcal {H}) denote the algebra of all bounded linear operators acting on H \mathcal {H} . Given a unitarily-invariant norm ‖ ⋅ ‖ u \| \cdot \|_u on B ( H ) \mathcal {B}(\mathcal {H}) and two linear operators A A and B B in B ( H ) \mathcal {B}(\mathcal {H}) , we shall say that A A and B B are polynomially isometric relative to ‖ ⋅ ‖ u \| \cdot \|_u if ‖ p ( A ) ‖ u = ‖ p ( B ) ‖ u \| p(A) \|_u = \| p(B) \|_u for all polynomials p p . In this paper, we examine to what extent an operator A A being polynomially isometric to a normal operator N N implies that A A is itself normal. More explicitly, we first show that if ‖ ⋅ ‖ u \| \cdot \|_u is any unitarily-invariant norm on M n ( C ) \mathbb {M}_n(\mathbb {C}) , if A , N ∈ M n ( C ) A, N \in \mathbb {M}_n(\mathbb {C}) are polynomially isometric and N N is normal, then A A is normal. We then extend this result to the infinite-dimensional setting by showing that if A , N ∈ B ( H ) A, N \in \mathcal {B}(\mathcal {H}) are polynomially isometric relative to the operator norm and N N is a normal operator whose spectrum neither disconnects the plane nor has interior, then A A is normal, while if the spectrum of N N is not of this form, then there always exists a nonnormal operator B B such that B B and N N are polynomially isometric. Finally, we show that if A A and N N are compact operators with N N normal, and if A A and N N are polynomially isometric with respect to the ( c , p ) (c,p) -norm studied by Chan, Li, and Tu, then A A is again normal.

We investigate systems of the form { A t g : g ∈ G , t ∈ [ 0 , L ] } where A ∈ B ( H ) is a normal operator in a separable Hilbert space H , G ⊂ H is a countable set, and L is a positive real number. Although the main goal of this work is to study the frame properties of { A t g : g ∈ G , t ∈ [ 0 , L ] } , as intermediate steps, we explore the completeness and Bessel properties of such systems from a theoretical perspective, which are of interest by themselves. Beside the theoretical appeal of investigating such systems, their connections to dynamical and mobile sampling make them fundamental for understanding and solving several major problems in engineering and science.

Given the abstract evolution equation \[ y'(t)=Ay(t),\ t\ge 0, \] with scalar type spectral operator $A$ in a complex Banach space, found are conditions necessary and sufficient for all weak solutions of the equation, which a priori need not be strongly differentiable, to be strongly Gevrey ultradifferentiable of order $\beta\ge 1$, in particular analytic or entire, on the open semi-axis $(0,\infty)$. Also, revealed is a certain interesting inherent smoothness improvement effect., Comment: Rewritten abstract, minor corrections and readability improvements, updated bibliography. arXiv admin note: substantial text overlap with arXiv:1706.08014

In this paper, we investigate the question of when the equations A ⁎ s A s = ( A ⁎ A ) s , s ∈ S , where S is a finite set of positive integers, imply the quasinormality or normality of A. In particular, it is proved that if S = { p , m , m + p , n , n + p } , where p ⩾ 1 and 2 ⩽ m n , then A is quasinormal. Moreover, if A is invertible and S = { m , n , n + m } with m ⩽ n , then A is normal. The case when S = { m , m + n } and A ⁎ n A n ⩽ ( A ⁎ A ) n is also discussed.

Nonlinear amplifiers such as the transistor are ubiquitous in classical technology, but their quantum analogues are not well understood. We introduce a class of nonlinear amplifiers that amplify any normal operator and add only a half-quantum of vacuum noise at the output. In the large-gain limit, when used in conjunction with a noisy linear detector, these amplifiers implement ideal measurements of the normal operator., Comment: 8 pages + appendix. V2 ~ published version, including a new section on three mode amplifiers for measuring normal operators

In this note, we consider properties of a compact normal operator A such that ‖ A ‖ ⩽ ‖ A + D ‖ for all D ∈ D ( K ( H ) ) , or equivalently, ‖ A ‖ = dist ( A , D ( K ( H ) ) ) , where ‖ ⋅ ‖ denotes the usual operator norm and D ( K ( H ) ) is the subalgebra of diagonal compact operators in a fixed orthonormal basis of the Hilbert space H .

This paper deals with a conjecture posed by Abbas Salemi in 2011 (Banach J. Math. Anal.), claiming that for the spectrum of every bounded linear operator on a separable Hilbert space there exists a...

The dynamics of the eigenvalues of a family of Sturm–Liouville operators with complex integrable PT-symmetric potential on a finite interval is studied. In the model case of the complex Airy operator, a criterion for the similarity of operators in the family to self-adjoint and normal operators is stated and the exceptional parameter values corresponding to multiple eigenvalues are analytically calculated.

Some stronger and equivalent metrics are defined on $\mathcal{M}$, the set of all bounded normal operators on a Hilbert space $H$ and then some topological properties of $\mathcal{M}$ are investigated.

In this note we obtain commutativity criteria for normal compact operators using the projective spectrum. We thus improve a corresponding result obtained by Chagouel, Stessin and Zhu in Trans. Amer. Math. Soc. 368 (2016), 1559–1582.

We investigate several properties of Aluthge transform of an operator . We prove (i) if T is (s, p)-w-hyponormal operator and is quasinormal (resp., normal), then T is quasinormal (resp., normal), (ii) if T is (s, p)-w-hyponormal operator and is a partial isometry, then T is quasinormal partial isometry, (iii) if T and are (s, p)-w-hyponormal operator, then T is normal, and (iv) Fuglede–Putnam type theorem holds for a class p-w-hyponormal operator T with if T satisfies a kernel condition

Finite rank perturbations $T=N+K$ of a bounded normal operator $N$ on a separable Hilbert space are studied thanks to a natural functional model of $T$; in its turn the functional model solely relies on a perturbation matrix/ characteristic function previously defined by the second author. Function theoretic features of this perturbation matrix encode in a closed-form the spectral behavior of $T$. Under mild geometric conditions on the spectral measure of $N$ and some smoothness constraints on $K$ we show that the operator $T$ admits invariant subspaces, or even it is decomposable., Comment: 33 pages; to appear in Journal of Operator Theory

Given the abstract evolution equation \[ y'(t)=Ay(t),\ t\in \mathbb{R}, \] with a scalar type spectral operator $A$ in a complex Banach space, we find conditions on $A$, formulated exclusively in terms of the location of its spectrum in the complex plane, necessary and sufficient for all weak solutions of the equation, which a priori need not be strongly differentiable, to be strongly Gevrey ultradifferentiable of order $\beta\ge 1$, in particular analytic or entire, on $\mathbb{R}$. We also reveal certain inherent smoothness improvement effects and show that, if all weak solutions of the equation are Gevrey ultradifferentiable of orders less than one, then the operator $A$ is necessarily bounded. The important particular case of the equation with a normal operator $A$ in a complex Hilbert space follows immediately., Comment: Minor readability improvement. arXiv admin note: substantial text overlap with arXiv:1707.09359, arXiv:1706.08014, arXiv:1803.10038, arXiv:1708.05067

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.

In this paper we investigate the asymptotic behaviour of singular numbers of the operator P h C P h , where C is Cauchy's operator and P h is the orthogonal projection from L 2 ( D ) onto harmonic function subspace. We prove that s n ( P h C P h ) = O ( 1 n ) , as n → + ∞ . Moreover, we find the upper and lower asymptotic estimates, π − 1 ⩽ lim n → + ∞ ⁡ n s n ( P h C P h ) ⩽ π − 1 ( 35 + 21 2 6 ) + 7 6 .

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 $\|A\sharp C\|\geq\|B\|$ .

In this paper, we give a characterization of a complex symmetric Toeplitz operator T φ on the Hardy space H 2 . Moreover, if T φ is a complex symmetric Toeplitz operator, we provide a necessary and sufficient condition for T φ to be normal. Finally, we investigate these T φ with finite symbols.

In this paper we introduce a modified spectral method for solving the linear operator equation L u = f , L : D ( L ) ? H 1 ? H 2 , where H 1 and H 2 are normed vector spaces with norms ? . ? 1 and ? . ? , respectively and D ( L ) is the domain of L . Also for each h ? H 2 , ? h ? 2 = ( h , h ) where ( . , . ) is an inner product on H 2 . In this method we make a new set { ? n } n = 0 ∞ for H 1 using L and two sets in H 1 and H 2 . Then using the new set { ? n } n = 0 ∞ we solve this linear operator equation. We show that this method does not have some shortcomings of spectral method, also we prove the stability and convergence of the new method. After introducing the method we give some conditions that under them the nonlinear operator equation L u + N u = f can be solved. Some examples are considered to show the efficiency of method.

We prove that an injective, not necessarily bounded weighted bilateral shift operator on $\ell^2(\mathbb{Z})$ is similar to a normal operator if and only if it is similar to a scalar multiple of the simple (i.e. unweighted) bilateral shift operator $S$., Comment: 6 pages, Keywords: Bilateral weighted shift operator, similarity to normal operators

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.

We characterize normal Toeplitz operator on the Fock spaces F2(C). First, we state basic properties for Toeplitz operator T&phi, on F2(C). Next, we study the normal Toeplitz operator T&phi, on F2(C) in terms of harmonic symbols &phi, Finally, we characterize the normal Toeplitz operators T&phi, with non-harmonic symbols acting on F2(C).

In this paper, we give a characterization of normality of Toeplitz operator T&phi, on the Bergman space A2(D). First, we state basic properties for Toeplitz operator T&phi, on A2(D). Next, we consider the normal Toeplitz operator T&phi, on A2(D) in terms of harmonic symbols &phi, Finally, we characterize the normal Toeplitz operators T&phi, with non-harmonic symbols acting on A2(D).

We show that the normal operator of the X-ray transform in $\mathbb{R}^d$, $d\geq 2$, has a unique continuation property in the class of compactly supported distributions. This immediately implies uniqueness for the X-ray tomography problem with partial data and generalizes some earlier results to higher dimensions. Our proof also gives a unique continuation property for certain Riesz potentials in the space of rapidly decreasing distributions. We present applications to local and global seismology. These include linearized travel time tomography with half-local data and global tomography based on shear wave splitting in a weakly anisotropic elastic medium., 21 pages, 4 figures

Let H be a right quaternionic Hilbert space and let T be a quaternionic normal operator with domain D ( T ) ⊂ H . We prove that there exists a Hilbert basis N of H , a measure space ( Ω 0 , ν ) , a unitary operator U : H → L 2 ( Ω 0 ; H ; ν ) and a ν-measurable function η : Ω 0 → C such that T x = U ⁎ M η U x , for all x ∈ D ( T ) where M η is the multiplication operator on L 2 ( Ω 0 ; H ; ν ) induced by η with U ( D ( T ) ) ⊆ D ( M η ) . We show that every complex Hilbert space can be seen as a slice Hilbert space of some quaternionic Hilbert space and establish the main result by reducing the problem to the complex case then lift it to the quaternion case.

In this paper, we obtain numerical ranges of normal weighted composition operators on $\ell ^{2}(\mathbb {N})$.

We present a new method for region of interest (ROI) reconstruction from high-resolution, nontruncated data in parallel-ray X-ray computed tomography (CT). Many of the approaches to ROI CT reconstruction in the literature rely on a costly forward projection step to form an ROI-only sinogram. Our approach instead relies on a digital filtering implementation of the normal operator ((HH)-H-T) to compute a back projected version of the ROI-only sinogram that can be used directly for reconstruction, thus eliminating the forward projection step altogether. Results on three synthetic datasets with a variety of experimental conditions show that the method provides accuracy on par with a full reconstruction at a fraction of the time and memory cost.

A simple proof is provided to show that any bounded normal operator on a real Hilbert space is orthogonally equivalent to its transpose(adjoint). A structure theorem for invertible skew-symmetric operators, which is analogous to the finite dimensional situation is also proved using elementary techniques. The second result is used to establish the main theorem of this article, which is a generalization of Williamson's normal form for bounded positive operators on infinite dimensional separable Hilbert spaces. This has applications in the study of infinite mode Gaussian states., Comment: 16 pages. Minor improvements from previous version

The primary purpose of this paper is to investigate the question of invertibility of the sum of operators. The setting is bounded and unbounded linear operators. Some interesting examples and consequences are given. As an illustrative point, we characterize invertibility for the class of normal operators. Also, we give a very short proof of the self-adjointness of a normal operator when the latter has a real spectrum., Comment: 11 pages

Let $A$ be an operator on {a separable } Hilbert space $\cH$, and let $G \subset \cH$. It is known that - under appropriate conditions on $A$ and $G$ - the set of iterations $F_G(A)= \{A^j \gbf \; | \; \gbf \in G, \; 0 \leq j \leq L(\gbf) \} $ is a frame for $\cH$. We call $F_G(A)$ a dynamical frame for $\cH$, and explore further its properties; in particular, we show that the canonical dual frame of $F_G(A)$ also has an iterative set structure. We explore the relations between the operator $A$, the set $G$ and the number of iterations $L$ which ensure that the system $F_G(A)$ is a scalable frame. We give a general statement on frame scalability, We and study in detail the case when $A$ is a normal operator, utilizing the unitary diagonalization in finite dimensions. In addition, we answer the question of when $F_G(A)$ is a scalable frame in several special cases involving block-diagonal and companion operators.

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).

We consider the Schur-Horn problem for normal operators in von Neumann algebras, which is the problem of characterizing the possible diagonal values of a given normal operator based on its spectral data. For normal matrices, this problem is well-known to be extremely difficult, and in fact, it remains open for matrices of size greater than $3$. We show that the infinite dimensional version of this problem is more tractable, and establish approximate solutions for normal operators in von Neumann factors of type I$_\infty$, II and III. A key result is an approximation theorem that can be seen as an approximate multivariate analogue of Kadison's Carpenter Theorem., 34 pages

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.