On operators preserving the numerical range (original) (raw)
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Linear operators preserving the decomposable numerical range
Linear and Multilinear Algebra, 1979
Let 1 ≤ m ≤ n, and let χ : H → C be a degree 1 character on a subgroup H of the symmetric group of degree m. The generalized matrix function on an m × σ(j) , and the decomposable numerical radius of an n × n matrix A on orthonormal tensors associated with χ is defined by r ⊥ χ (A) = max{|d χ (X * AX)| : X is an n × m matrix such that X * X = I m }.
Linear maps on ℬ(ℋ) preserving some operator properties
Proyecciones (Antofagasta), 2021
In this paper, for a complex Hilbert space ℋ with dim ℋ ≥ 2, we study the linear maps on ℬ(ℋ), the bounded linear operators on ℋ, that preserves projections and idempotents. As a result, we characterize the linear maps on ℬ(ℋ) that preserves involutions in both directions.
2017
Let be a Hilbert space equipped with the inner product , and let be the algebra of bounded linear operators acting on . We recall that the numerical range (also known as the field of values) of is the collection of all complex numbers of the form where is a unit vector in . i.e. See, ([2], [5], [8]) which is useful for studying operators. In particular, the geometrical properties of the numerical range often provide useful information about the algebraic and analytic properties of the operator . For instance, if and only if ; is real if and only if , has no interior points if and only if there are complex numbers, and with such that is self-adjoint. Moreover, the closure of denoted by , always contains the spectrum of denoted by . See, [8] Let denote the set of compact operators on and be the canonical quotient map. The essential numerical range of , denoted by is the set; See, ([1], [2], [3]) where the intersection runs over the compact operators . Chacon and Chacon [3] gave some o...
Range Kernel Orthogonality and Finite Operators
Kyungpook mathematical journal, 2015
Let H be a separable infinite dimensional complex Hilbert space, and let L(H) denote the algebra of all bounded linear operators on H into itself. Let A, B ∈ L(H) we define the generalized derivation δA,B : L(H) → L(H) by δA,B(X) = AX − XB, we note δA,A = δA. If the inequality ||T − (AX − XA)|| ≥ ||T || holds for all X ∈ L(H) and for all T ∈ kerδA, then we say that the range of δA is orthogonal to the kernel of δA in the sense of Birkhoff. The operator A ∈ L(H) is said to be finite [22] if ||I − (AX − XA)|| ≥ 1(*) for all X ∈ L(H), where I is the identity operator. The well-known inequality (*), due to J. P. Williams [22] is the starting point of the topic of commutator approximation (a topic which has its roots in quantum theory [23]). In [16], the author showed that a paranormal operator is finite. In this paper we present some new classes of finite operators containing the class of paranormal operators and we prove that the range of a generalized derivation is orthogonal to its kernel for a large class of operators containing the class of normal operators.
Condition for the higher rank numerical range to be non-empty, Linear and Multilinear Algebra
It is shown that the rank-k numerical range of every n-by-n complex matrix is non-empty if k < n/3+1. The proof is based on a recent characterization of the rank-k numerical range by Li and Sze, the Helly's theorem on compact convex sets, and some eigenvalue inequalities. In particular, the result implies that Λ 2 (A) is non-empty if n ≥ 4. This confirms a conjecture of Choi et al. If k ≥ n/3 + 1, an n-by-n complex matrix is given for which the rank-k numerical range is empty. Extension of the result to bounded linear operators acting on an infinite dimensional Hilbert space is also discussed.
Linear mappings that preserve potent operators
Proceedings of the American Mathematical Society, 1995
Let H and Í bea complex Hubert spaces, while 3S(H) and ¿&(K) denote the algebras of all linear bounded operators on H and K , respectively. We characterize surjective linear mappings from ¿&(H) onto 3 §(K) that preserve potent operators in both directions.
Linear maps preserving G-unitary operators in Hilbert space
Arab Journal of Mathematical Sciences, 2015
Let H be a complex Hilbert space and BðHÞ the algebra of all bounded linear operators on H. We give the concrete forms of surjective continuous unital linear maps from BðHÞ onto itself that preserve G-unitary operators.