Let H be an infinite-dimensional separable complex Hilbert space and \mathcal{B}(H) the algebra of all bounded linear operators on H. In this paper we characterize surjective linear maps \phi : ...F\mathcal{B}(H)\to \mathcal{B}(H) preserving the set of Fredholm operators in both directions. As an application we prove that \phi preserves the essential spectrum if and only if the ideal of all compact operators is invariant under \phi and the induced linear map \varphi on the Calkin algebra is either an automorphism, or an anti-automorphism. Moreover, we have, either ind(\phi(T)) = ind(T) or ind(\phi(T)) = - ind(T) for every Fredholm operator T.
Let H be a complex Hilbert space and let B(H) be the algebra of all bounded linear operators on H. The polar decomposition theorem asserts that every operator T∈B(H) can be written as the product ...T=VP of a partial isometry V∈B(H) and a positive operator P∈B(H) such that the kernels of V and P coincide. Then this decomposition is unique. V is called the polar factor of T. Moreover, we have automatically P=|T|=(T⁎T)12. Unlike P, we do not have any representation formula for V. In this paper, we give several explicit formulas representing the polar factor. These formulas allow for methods of approximations of the polar factor of T.
Let B(H) be the algebra of all bounded linear operators on a complex Hilbert space H. For any operator T∈B(H) and unit vector x∈H, let γ(T,x) denote the local reduced minimum modulus of T at x. In ...this paper, we characterize surjective maps on B(H) such that, for all S,T∈B(H) and all unit vectors x∈H, one hasγ(T−S,x)=0 if and only if γ(Φ(T)−Φ(S),x)=0. We also describe the form of all maps on B(H) preserving the product of operators of zero local reduced minimum moduli. Furthermore, some consequences and open problems are also discussed.
Let B(H) be the algebra of all bounded linear operators acting on a Hilbert space H. The main purpose in this paper is to obtain a characterization of bijective maps Φ:B(H)→B(K), K Hilbert space, ...satisfying the following conditionΔλ(Φ(A)∘Φ(B))=Φ(Δλ(A∘B)) for all A,B∈B(H), where Δλ(T) stands the λ-Aluthge transform of the operator T∈B(H) and A∘B=12(AB+BA) is the Jordan product of A and B. We prove that a bijective map Φ satisfies the above condition, if and only if there exists a unitary linear bounded operator U:H→K, such that Φ has the form Φ(A)=UAU⁎ for all A∈B(H).
Let n be an integer greater than 1, and Mn(C) be the algebra of all n×n-complex matrices. Let x0∈Cn be a nonzero vector, and Φ be a linear map on Mn(C) such that Φ(I) is invertible. For any matrix ...T∈Mn(C), let γ(T,x0) denote the local reduced minimum modulus of T at x0. In this paper, we show that Φ satisfiesγ(T,x0)=0⇔γ(Φ(T),x0)=0,(T∈Mn(C)), if and only if there are two invertible matrices A,B∈Mn(C) such that Ax0=A⁎x0=x0 and Φ(T)=BTA for all T∈Mn(C). When n=2, we show that the invertibility hypothesis of Φ(I) is redundant.
In this paper, we give a complete form of bijective (not necessarily linear) maps
where H,K are Hilbert spaces with
that satisfy
where
is the λ-Aluthge transform of T and the operation
is the
...-Jordan-triple product. We show that there exists a unitary or anti-unitary operator
and a constant
, with
such that
In this paper we give several expressions of spectral radius of a bounded operator on a Hilbert space, in terms of iterated λ-Aluthge transform, numerical radius and the asymptotic behavior of the ...powers of this operator.
We also obtain several characterizations of normaloid operators.
We investigate some bounded linear operators T on a Hilbert space which satisfy the condition |T|≤|ReT|. We describe the maximum invariant subspace for a contraction T on which T is a partial ...isometry to obtain that, in certain cases, the above condition ensures that T is self-adjoint. In other words we show that the Fong–Tsui conjecture holds for partial isometries, contractive quasi-isometries, or 2-quasi-isometries, and Brownian isometries of positive covariance, or even for a more general class of operators.
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