Search

In this paper, further characterizations of selfadjoint operators on Hilbert spaces are established. Thus the recent results concerning two-sided removal and cancellation laws associated with a Hermitian matrix (Central University of Finance and Economics, DOI: https://doi.org/10.21203/rs.3.rs-1104901/v1) are extended to the infinite dimensional setting with proofs based on operator matrices. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of (A , m) -partial isometry for a positive operator A and a nonnegative integer m. This family of operators contains both the class of (A , m) -isometries discussed in Sid Ahmed and Saddi [A-m-isometric operators in semi-Hilbertian spaces. Linear Algebra Appl. 2012;436:3930–3942] and that of m-partial isometries introduced in Saddi and Sid Ahmed [m-partial isometries on Hilbert spaces. Int J Funct Anal Oper Theory Appl. 2010;2(1):67–83]. First, we give some interesting algebraic properties of (A , m) -partial isometries, then we discuss a necessary and sufficient condition for an (A , m) -partial isometry to be an (A , m) -isometry. Finally, we give some spectral properties of (A , m) -partial isometries. [ABSTRACT FROM AUTHOR]

This paper is concerned with stability of deficiency indices of Hermitian subspaces (i.e. linear relations) under relatively bounded perturbations in Hilbert spaces. Several results about invariance of deficiency indices of Hermitian subspaces under relatively bounded perturbations are established. As a consequence, invariance of self-adjointness of Hermitian subspaces under relatively bounded perturbations is obtained. In addition, it is shown that the deficiency indices may shrink in the special case that the relative bound is equal to 1. The results obtained in the present paper generalize the corresponding results for symmetric operators to more general Hermitian subspaces, some of which relax or improve certain conditions of the related results in existing literatures. [ABSTRACT FROM AUTHOR]

The paper deals with the generalized numerical radius of linear operators acting on a complex Hilbert space H , which are bounded with respect to the seminorm induced by a positive operator A on H . Here A is not assumed to be invertible. Mainly, if we denote by ω A (⋅) and ω (⋅) the generalized and the classical numerical radii respectively, we prove that for every A-bounded operator T we have ω A (T) = ω ( A 1 / 2 T (A 1 / 2 ) † ) , where (A 1 / 2 ) † is the Moore-Penrose inverse of A 1 / 2 . In addition, several new inequalities involving ω A (⋅) for single and several operators are established. In particular, by using new techniques, we cover and improve some recent results due to Najafi [Linear Algebra Appl. 2020;588:489–496]. [ABSTRACT FROM AUTHOR]

Let ( X , ⟨ ⋅ , ⋅ ⟩) be a Hilbert C ∗ -module over a C ∗ -algebra A and let S (A) be the set of states on A . In this paper, we first compute the norm derivative for nonzero elements x and y of X as follows: lim t → 0 + ‖ x + t y ‖ − ‖ x ‖ t = 1 ‖ x ‖ max { Re φ (⟨ x , y ⟩) : φ ∈ S (A) , φ (⟨ x , x ⟩) = ‖ x ‖ 2 }. We then apply it to characterize different concepts of orthogonality in X . In particular, we present a simpler proof of the classical characterization of Birkhoff–James orthogonality in Hilbert C ∗ -modules. Moreover, some generalized Daugavet equation in the C ∗ -algebra B (H) of all bounded linear operators acting on a Hilbert space H is solved. [ABSTRACT FROM AUTHOR]

The concept of the Drazin inverse of multivalued operators in a Banach space studied by A. Ghorbel and M. Mnif [Monatsh Math. 2019;189:273–293] is generalized in the context of the generalized Drazin inverse of multivalued operators [Rocky Mountain J Math. 2020;50(4):1387–1408]. The purpose of this paper is to present new additive results for this concept. In particular, we give a sufficient condition for an everywhere defined linear relation to have at most one Drazin inverse. Some properties and the explicit expressions for the Drazin inverse of the product are obtained. Also, some results of C. Deng, H. Du [Proc Amer Math Soc. 2006;134:3309–3317] concerning the reduced minimum modulus of Drazin inverses of linear operators on Hilbert spaces are extended to the case of linear relations. [ABSTRACT FROM AUTHOR]

We address a general question whether geometry of submanifolds of an integral compact Kähler manifold is characterized by invariants that come from analysis. In geometric quantization, we have an integral compact Kähler manifold M and a holomorphic line bundle L on this manifold. There is a known procedure how to associate a sequence of mixed states (ρ N) , N = 1 , 2 , 3 , ... , to a submanifold Λ of M. Do analytic properties of this sequence reflect the geometry of Λ ? In this paper, we consider the case when M is a product of two integral compact Kähler manifolds. We show that, when Λ is a product submanifold of M, then the entanglement of formation of ρ N is zero for all sufficiently large N. [ABSTRACT FROM AUTHOR]

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 equal. We also show that the spherical spectrum of a quaternionic bounded operator is included in the closure of its numerical range, and we show that the numerical range of an operator on a quaternionic Hilbert space is not necessarily convex. For a quaternionic bounded normal operator, we prove that the convex hull of the closure of its numerical range is equal to the convex hull of its spherical spectrum. Finally, we give some inequalities between the numerical radius, the spectral radius and the norm of a right linear bounded operator, and we prove also that the norm and the numerical radius of a quaternionic bounded hyponormal operator are equal. [ABSTRACT FROM AUTHOR]

In this paper, some sufficient conditions and necessary conditions are established for existence of nonzero solutions of the operator equation XAX = BX on Hilbert spaces. Moreover, the necessary and sufficient condition which the solution of the equation is idempotent is also presented. [ABSTRACT FROM AUTHOR]

Let H be a complex separable infinite-dimensional Hilbert space. Given the operators A ∈ B(H) and B ∈ B(H), we define MX := ... where X ∈ S(H) is a self-adjoint operator. In this paper, a necessary and sufficient condition is given for MX to be a left (right) invertible operator for some X ∈ S(H). Moreover, it is shown that ..., where σ* is the left (right) spectrum. Finally, we further characterize the perturbation of the left (right) spectrum for Hamiltonian operators. [ABSTRACT FROM AUTHOR]

This paper is concerned with the lower and upper redundancy of infinite frames in a separable Hilbert space. Using unit norm linear operators, a new representation of upper redundancy is given. Also, we show that there is no frame such that its lower redundancy is less than one and its upper redundancy is one. Moreover, for given (μ , λ) ∈ M := { (μ , λ) ; 1 ≤ ⌈ μ ⌉ ≤ ⌊ λ ⌋ } ∖ { (μ , 1) ; 0 < μ < 1 } , we construct a frame such that its lower redundancy is μ and its upper redundancy is λ. [ABSTRACT FROM AUTHOR]

Let (H, 〈 · , · 〉) be a complex Hilbert space and A be a positive bounded linear operator on H. The semi-inner product 〈x, y〉A: = 〈Ax, y〉, x, y ∈ H, induces a semi-norm ∥ ⋅ ∥ A on H. Let ωA(T) and ∥ T ∥ A denote the A-numerical radius and the A-operator semi-norm of an operator T in semi-Hilbertian space (H, 〈 · , · 〉A), respectively. In this paper, some new bounds for the A-numerical radius of operators in semi-Hilbertian space are obtained, which improve the existing ones. In particular, a refinement of the triangle inequality for A-operator semi-norm is also shown. [ABSTRACT FROM AUTHOR]

In this paper, we obtain some inequalities for the Berezin number of operators on reproducing kernel Hilbert spaces by using the Krein–Lin inequality and refinements of the Young inequality. Also, we give upper bounds for b e r 2 (A) − b e r (A 2) on reproducing kernel Hilbert spaces. [ABSTRACT FROM AUTHOR]

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 M 2 , A , B on the class of Hilbert–Schmidt operators by M 2 , A , B (X) = A X B. In this paper, we show that if B is normal, then c o (W 0 (A) W 0 (B)) ⊆ W 0 (M 2 , A , B) , where co stands for the convex hull and W 0 (.) denotes the maximal numerical range. If in addition, A is hyponormal, this inclusion becomes an equality. Some remarks about the maximal numerical range of the generalized derivation δ 2 , A , B on the class of Hilbert–Schmidt operators are also given. [ABSTRACT FROM AUTHOR]

In this paper, we first investigate some inequalities involving the p-weighted geometric operator mean A ♯ p B = A 1 / 2 ( A − 1 / 2 B A − 1 / 2 ) p A 1 / 2 , where p ∈ [0, 1] is a real number and A, B are two positive invertible operators acting on a Hilbert space. As applications, we obtain some inequalities about the so-called Tsallis relative operator entropy. We also give some inequalities involving the Heinz operator mean. Our results refine some inequalities existing in the literature. In a second part, we construct iterative algorithms converging to A ♯ p B with a high rate of convergence. Some relationships involving A ♯ p B are deduced. Numerical examples illustrating the theoretical results are also discussed. [ABSTRACT FROM AUTHOR]

In this paper, we extend the operations of parallel addition A:B and parallel subtraction A ÷ B from the cone of bounded nonnegative self-adjoint operators to the linear bounded operators on a Hilbert space. Some conditions for the relations A † : B † = (A + B) † , B = (A : B) ÷ A , (A C) ÷ (B C) = (A ÷ B) C , A ÷ B = (P (A † − B †) P) † , B = A : (B ÷ A) , (C A) : (C B) = C (A : B) to be true are studied and the solution of the equation A:X = B is investigated. Moreover, some relationships between the parallel addition and subtraction of projections are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we consider a generalization of (A , m) -isometric Hilbert space operators to the multivariable setting. Inspired by the work [Sid Ahmed OAM, Chō M, Lee JE. On (m,C)-isometric commuting tuples of operators on a Hilbert space. Res Math. 2018;73:51. Doi:], we introduce the class of (A , m) -isometric tuples of commuting operators. A d-tuple T = (T 1 , ... , T d) ∈ L (H) d is said to be an (A , m) -isometric tuple of operators if ∑ k = 0 m (− 1) m − k m k ∑ | α | = k k ! α ! T ∗ α A T α = 0 for some positive integer m and some positive operator A. We study some basic properties of these tuples of commuting operators which generalize those established in Gu [Exapmles of m-isometric tuples of operators on a Hilbert space. J Korean Math Soc. 2018;55(1):225–251], Gleason and Richter [m-Isometric commuting tuples of operators on a Hilbert space. Int Equ Oper Theory. 2006;56(2):181–196], and Sid Ahmed and Saddi [A-m-Isometric operators in semi-Hilbertian spaces. Linear Algebra Appl. 2012;436(10): 3930–3942]. [ABSTRACT FROM AUTHOR]

In this paper, we discuss a version of Birkhoff–James orthogonality in the C ∗ -algebra of all bounded linear operators on a finite-dimensional Hilbert space. We obtain a characterization of this relation and use it to describe some classes of operators. We also characterize linear preservers of this type of orthogonality. [ABSTRACT FROM AUTHOR]

Let N be a nest on a complex separable Hilbert space H and A l g N be the associated nest algebra. In this paper, we prove that every bilocal Lie derivation from A l g N into itself is of the form A → [ A , T ] + λ A + f (A) , where T ∈ A l g N , λ ∈ C and f : A l g N → C I is a linear map vanishing on each commutator. Moreover, we show that every bilocal Lie derivation from A l g N into itself is a Lie derivation if N is a non-atomic nest or there exists an atom E of N with dim ⁡ E > 1. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the invertibility of generalized g-Bessel multipliers. Sufficient and necessary conditions for invertibility are determined depending on the optimal g-frame bounds. Moreover, we show that, for semi-normalized symbols, the inverse of any invertible generalized g-frame multiplier can be represented as a generalized g-frame multiplier with the reciprocal symbol and dual g-frames of the given ones. Furthermore, we investigate some equivalent conditions for the special case, when both dual g-frames can be chosen to be the canonical duals. Finally, we give several approaches for constructing invertible generalized g-frame multipliers from the given ones. It is worth mentioning that some of our results are quite different from those studied in the previous literatures on this topic. [ABSTRACT FROM AUTHOR]

Let H and K be infinite dimensional separable Hilbert spaces. For A ∈ L R (H) , B ∈ L R (K) and C ∈ L R (K , H) , we denote by M C = A C 0 B the upper triangular relation matrix. In this paper, the sets ⋂ C ∈ C M (K , H) σ (M C) , ⋂ C ∈ L R (K , H) σ 1 (M C) and ⋂ C ∈ B (K , H) σ 2 (M C) are characterized, where σ 1 ∈ { σ p , σ r , σ c } and σ 2 ∈ { σ , σ p , σ r , σ c }. Moreover, the relationship between σ (M C) , σ (A) and σ (B) is described for given relations A ∈ B C R (H) , B ∈ B C R (K) and C ∈ B R (K , H) under the local spectral theory. [ABSTRACT FROM AUTHOR]

In this paper, we establish some numerical radius inequalities for 2 × 2 bounded linear operator defined on a complex Hilbert space. As a natural application, the existence of the all polynomial zeros is identified in a specific small disk. Moreover, we provide a refinement of an earlier numerical radius inequality due to Herzallah et al. [Numerical radius inequalities for certain 2 × 2 operator matrices. Integr Equ Oper Theory. 2011;71:129–147] and a generalization of Shebrawi's inequality [Numerical radius inequalities for certain 2 × 2 operator matrices II. Linear Algebra Appl. 2017;523:1–12]. [ABSTRACT FROM AUTHOR]

In this paper, we study the dilations of frame generators and pairs of dual frame generators of two special structured unitary systems for Hilbert spaces, i.e. the product unitary system and group-like unitary system. We first introduce certain operators which are associated with the frame generators of the unitary system and some properties of the operators are established. Then, as an application, we show that the canonical dual of a frame of group-like unitary systems remain the group-like structure. And we use these properties to study the dilations of the frame generators and dual pairs of frame generators of these two types of unitary systems. [ABSTRACT FROM AUTHOR]

The issue of the S-spectrum introduced by Colombo et al. [Noncommutative functional calculus. Theory and applications of slice hyperholomorphic functions. Basel: Birkhäuser/Springer Basel AG; 2011. (Progress in mathematics; vol. 289). p. vi+221]. motivates us to investigate the pseudo S-spectrum of a bounded right quaternionic linear operator, in a right quaternionic Hilbert space with a left multiplication defined on it. Besides, we give a characterization for the Weyl pseudo S-spectrum in terms of the Fredholm operators with the aid of results from the papers [Muraleetharan, Thirulogasanthar. Fredholm operators and essential S-spectrum in the quaternionic setting. J Math Phys. 2018;59(10):103506, 27 pp.; Muraleetharan, Thirulogasanthar. Weyl and Browder S-spectra in a right quaternionic Hilbert space. J Geom Phys. 2019;135:7–20]. [ABSTRACT FROM AUTHOR]

In this paper, we give new results for the p-numerical radii w p ⋅ of Hilbert space operators. It is shown, among other inequalities, that if A is a Hilbert space operator, which belongs to the Schatten p-class, then w p p (A) ≥ w p / 2 p / 2 (A 2) 2 p / 2 + A ∗ A + A A ∗ p / 2 p / 2 2 p + inf θ ∈ R Re ⁡ (e i θ A) p p − Im ⁡ (e i θ A) p p 2 and w p p (A) ≤ 2 p / 2 − 2 inf θ ∈ R Re ⁡ ((e i θ A) 2) p / 2 p / 2 + A ∗ A + A A ∗ p / 2 p / 2 4 for 4 ≤ p < ∞. Also, w p p (A) ≥ w p / 2 p / 2 A 2 4 + A ∗ A + A A ∗ p / 2 p / 2 2 p / 2 + 2 + inf θ ∈ R Re ⁡ (e i θ A) p p − Im ⁡ (e i θ A) p p 2 and w p p (A) ≤ inf θ ∈ R Re ⁡ ((e i θ A) 2) p / 2 p / 2 + A ∗ A + A A ∗ p / 2 p / 2 2 p / 2 for 2 ≤ p ≤ 4 , where ⋅ p is the Schatten p-norm. Applications of these inequalities to certain classes of operators are also given. [ABSTRACT FROM AUTHOR]

Let σ n (H) denote the class of operators A acting on a complex Hilbert space H such that W 0 (A) = c o (λ ∈ σ (A) : λ = A ) , where W 0 (A) and σ (A) are the maximal numericl range, the spectrum of A, respectively, and co stands for the convex hull. In this paper, we show that the class σ n (H) contains hyponormal operators. We also show that if two operators A ∈ σ n (H) and B ∈ σ n (H ′) have the same closure of the numerical range or have the same spectrum, then they also have the same maximal numerical range. An affirmative answer to the question 2.16 in Fialkow [Elementary operators and applications. Proceedings of the International Workshop; Bleuberen; 1991. p. 55–113] is also given. [ABSTRACT FROM AUTHOR]

Let H be a separable complex Hilbert space with dim H ≥ 3, B s (H) be the Lie algebra of all bounded self-adjoint operators on H, and let F : i B s (H) → [ d , ∞ ] with d ≥ 0 be a radial unitary similarity invariant function. In this paper, a structure feature is obtained for maps φ on B s (H) satisfying F (φ (A) φ (B) − φ (B) φ (A)) = F (A B − B A) for all A , B ∈ B s (H). As applications, we show that, for a surjective map φ on B s (H) , the following conditions are equivalent: φ preserves the p-norm for some 1 ≤ p < ∞ on Lie products; φ preserves the numerical radius on Lie products; φ preserves the pseudo-spectral radius on Lie products; there exists a unitary or conjugate unitary operator U on H, a sign function h : B s (H) → { 1 , − 1 } and a functional g : B s (H) → R such that φ (T) = h (T) U T U ∗ + g (T) I for all T ∈ B s (H). We also show that the following conditions are equivalent: φ preserves the numerical range on Lie products; φ preserves the pseudo spectrum on Lie products. Moreover, the concrete forms of the above preservers are given. The case dim ⁡ H = 2 is also discussed. [ABSTRACT FROM AUTHOR]

The purpose of the paper is to introduce and study a class of multivariable operators on semi-Hilbertian spaces, i.e. spaces generated by positive semidefinite sesquilinear forms. Let T = (T 1 , ... , T m) and let A be a commuting m-tuple of operators and a positive operator on a complex Hilbert space, respectively. We introduce an A-hyponormal tuple of operators and study some properties of this class of multivariable operators. An m-tuple of operators T = (T 1 , ... , T m) ∈ B A (H) m is said to be A-hyponormal tuple if ∑ 1 ≤ i , j ≤ m [ T i ♯ , T j ] x i | x j A ≥ 0 , for each finite collections (x i) 1 ≤ i ≤ m ∈ H. [ABSTRACT FROM AUTHOR]

A continuous function f : I × I → R is called operator Schur convex, if f is symmetric, namely f x , y = f y , x for all x, y ∈ I and f t A + 1 − t B , t B + 1 − t A ≤ f A , B in the operator order, for all A , B ∈ S A I H × S A I H and t ∈ 0 , 1 , where S A I H is the convex set of all selfadjoint operators on Hilbert space H with spectra in I. In this paper we investigate the main properties of such functions, establish some integral inequalities of Hermite–Hadamard, Čebyšev and Grüss' type and give some general classes of examples of operator Schur-convex functions. [ABSTRACT FROM AUTHOR]

This paper addresses continuous g-frames which are extensions of g-frames and continuous frames. First, we study the dual of continuous g-frames, which are critical components in reconstructions. We give the characterizations of dual continuous g-frames in terms of continuous g-preframe operators. The formulae of dual continuous g-frames for a given continuous g-frame are given. Second, we discuss the constructions of continuous g-frames in Hilbert spaces. We mainly consider the constructions of new continuous g-frames from known ones under certain conditions, which generalize the corresponding results on g-frames. In particular, we obtain a necessary and sufficient condition for the finite sum of continuous g-frames to be a continuous g-frame. [ABSTRACT FROM AUTHOR]

Let A be a factor von Neumann algebra acting on a complex separable Hilbert space H with dim (A) > 9. In this paper, we prove that every bilocal Lie derivation from A into itself is a Lie derivation. [ABSTRACT FROM AUTHOR]

This paper is concerned with the signal reconstruction without phase by norm retrievable frames in finite dimensional real Hilbert space H . Specifically, we show that the nonlinear analysis maps α , β : H ˆ ⟶ R m are injective, with α (x ˆ) := | x , f k | 1 ≤ k ≤ m and β (x ˆ) := | x , f k | 2 1 ≤ k ≤ m , where { f k } k = 1 m is a norm retrievable frame for H and H ˆ is the quotient space corresponding to a special equivalence relation on H . Using Householder matrices, the members of any x ˆ ∈ H ˆ are characterized. Also, we show that α has upper Lipschitz bound B and local lower Lipschitz bound A , with respect to an appropriate metric d on H ˆ , where A and B are the lower and upper frame bounds of the norm retrievable frame { f k } k = 1 m , respectively. Additionally, we show that β has upper Lipschitz bound 2 B and local lower Lipschitz bound a , with respect to an appropriate metric D on H ˆ , where a = inf ∥ e ∥ = 1 ∑ k = 1 m | e , f k | 4 and B is the upper frame bound of the norm retrievable frame { f k } k = 1 m . [ABSTRACT FROM AUTHOR]