Maps preserving numerical ranges of operator products

Abstract

Abstract. Let H be a complex Hilbert space, B(H) the algebra of all bounded linear operators on H and S a (H) the real linear space of all self-adjoint operators on H. We characterize the surjective maps on B(H) or S a (H) that preserve the numerical ranges of products or Jordan triple-products of operators.

Product of operators and numerical range preserving maps

“…For the usual product A 1 * · · · * A k = A 1 · · · A k and the Jordan triple product A * B = ABA, Hou and Di [5] have also obtained the result for B(H) in Theorem 1.1 with the surjectivity assumption. Our result is stronger when H is finite-dimensional.…”

“…In [2], it was shown that a multiplicative map φ : M n → M n satisfies W (φ(A)) = W (A) for all A ∈ M n if and only if φ has the form A → U * AU for some unitary matrix U ∈ M n . In [5], the authors replaced the condition that "φ is multiplicative and preserves the numerical range" on the surjective map φ : B(H) → B(H) by the condition that "W (AB) = W (φ(A)φ(B)) for all A, B", and showed that such a map has the form A → ±U * AU for some unitary operator U ∈ B(H). They also showed that a surjective map φ : B(H) → B(H) satisfies W (ABA) = W (φ(A)φ(B)φ(A)) for all A, B ∈ B(H) if and only if φ has the form A → µU * AU or A → µU * A t U for some unitary operator U ∈ B(H) and µ ∈ C with µ 3 = 1.…”

“…We thank Professor Jinchuan Hou for sending us the preprint [5], and his inspiring presentation of the paper at the workshop on preserver problems at the University of Hong Kong in 2004. Thanks are also due to Dr. Jor-Ting Chan for his effort in organizing the two workshops on preserver problems at the University of Hong Kong in 2004 and 2005 that facilitated this research.…”

MAPPING PRESERVING NUMERICAL RANGE OF OPERATOR PRODUCTS ON C*-ALGEBRAS

“…Therefore, we will concentrate our study henceforth on W . Recently, Hou and Di described in [9] surjective maps on the algebra B(H) which preserves the numerical range of the product. Namely, they characterized surjective mappings which satisfy one of the following conditions W (φ(a)φ(b)) = W (ab), (1a)…”

Algebra isomorphisms between standard operator algebras

“…[3,4,5,8,10,13]). Rather than requiring that such a map multiplicatively preserves the entire spectrum, however, it is also natural to ask whether preserving particular subsets of the spectrum will suffice.…”