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算子代数上保持ξ-Lie积的c-数值域的映射
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Abstract:
令B(H)是复Hilbert空间H上有界线性算子全体组成的代数，对于![\"\"](\"Edit_f3cfbee3-5437-447b-a62c-171d7c1e13bc.png\")
，Wc(A)表示算子A∈B(H)的c-数值域。本文主要研究了在H是有限维的情形下，B(H)上一类映射保持算子ξ-Lie积的数值域的刻画。具体说来，若![\"\"](\"Edit_424e63ec-3dc0-46b9-af1f-d32fc85d57d1.png\")
且c满足一定条件时，若![\"\"](\"Edit_c1f09ac4-2319-49cc-878f-a4915044d7e1.png\")
是满射，满足![\"\"](\"Edit_73b2519e-8517-4169-9067-959bf2d843ae.png\")
对任意的A,B∈B(H)成立，当且仅当存在H上的酉算子U以及常数![\"\"](\"Edit_da009416-ec53-4897-aae7-369313a12dae.png\")
使得![\"\"](\"Edit_7b58a7be-f8e3-42f8-8395-a577de1cfb1d.png\")
对所有T∈B(H)成立。
Let B(H) be the algebra of all bounded linear operators on an complex Hilbert space H. For ![\"\"](\"Edit_b40b0427-8b75-4285-8cae-e34b24929d1e.png\")
, Wc(A) denotes the c-numerical range of an operator A in B(H) . In this paper, we consider maps on B(H) prserving the c-numerical range of ξ-Lie Product. When the dimension of H is finite and ![\"\"](\"Edit_828309e5-d777-400a-ab89-476d4ccd5f54.png\")
belongs to a certain kind, it is shown that ![\"\"](\"Edit_da3edc62-6745-43a0-a30f-b7e1bf9c8e32.png\")
is surjective maps satisfying ![\"\"](\"Edit_bd79550e-9985-4bd2-9e5f-a087b5b8babd.png\")
for any A,B∈B(H) , ![\"\"](\"Edit_3cecae13-9d2e-42dc-b20e-7cf262db7f39.png\")
if and only if there exist a unitary operator U on H such that ![\"\"](\"Edit_6613306f-80d0-4eb4-aa20-ecbc8669e2cd.png\")
holds for all T∈B(H) , where ![\"\"](\"Edit_b2ea0272-94b6-4101-8a1b-e0668b0b4ea6.png\")
is a scalar.
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