Numerical radius preserving operators on
Author:
JorTing Chan
Journal:
Proc. Amer. Math. Soc. 123 (1995), 14371439
MSC:
Primary 47A12; Secondary 47B49
MathSciNet review:
1231293
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Abstract: Let H be a Hilbert space over and let denote the vector space of all bounded linear operators on H. We prove that a linear isomorphism is numerical radiuspreserving if and only if it is a multiply of a isomorphism by a scalar of modulus one.
 [1]
ChiKwong
Li, Linear operators preserving the
numerical radius of matrices, Proc. Amer. Math.
Soc. 99 (1987), no. 4, 601–608. MR 877024
(87m:15004), http://dx.doi.org/10.1090/S00029939198708770247
 [2]
ChiKwong
Li and NamKiu
Tsing, Linear operators that preserve the 𝑐numerical range
or radius of matrices, Linear and Multilinear Algebra
23 (1988), no. 1, 27–46. MR 943767
(89f:15020), http://dx.doi.org/10.1080/03081088808817854
 [3]
M.
Eidelheit, On isomorphisms of rings of linear operators,
Studia Math. 9 (1940), 97–105 (English, with
Ukrainian summary). MR 0003467
(2,224d)
 [4]
Paul
Richard Halmos, A Hilbert space problem book, 2nd ed.,
Graduate Texts in Mathematics, vol. 19, SpringerVerlag, New
YorkBerlin, 1982. Encyclopedia of Mathematics and its Applications, 17. MR 675952
(84e:47001)
 [5]
Ali
A. Jafarian and A.
R. Sourour, Spectrumpreserving linear maps, J. Funct. Anal.
66 (1986), no. 2, 255–261. MR 832991
(87m:47011), http://dx.doi.org/10.1016/00221236(86)90073X
 [6]
Richard
V. Kadison, Isometries of operator algebras, Ann. Of Math. (2)
54 (1951), 325–338. MR 0043392
(13,256a)
 [7]
Richard
V. Kadison, A generalized Schwarz inequality and algebraic
invariants for operator algebras, Ann. of Math. (2)
56 (1952), 494–503. MR 0051442
(14,481c)
 [8]
Richard
V. Kadison and John
R. Ringrose, Fundamentals of the theory of operator algebras. Vol.
I, Pure and Applied Mathematics, vol. 100, Academic Press, Inc.
[Harcourt Brace Jovanovich, Publishers], New York, 1983. Elementary theory.
MR 719020
(85j:46099)
 [9]
V.
J. Pellegrini, Numerical range preserving operators on a Banach
algebra, Studia Math. 54 (1975), no. 2,
143–147. MR 0388104
(52 #8941)
 [1]
 C. K. Li, Linear operators preserving the numerical radius of matrics, Proc. Amer. Math. Soc. 99 (1987), 601608. MR 877024 (87m:15004)
 [2]
 C. K. Li and N. K. Tsing, Linear operators that preserve the cnumerical range or radius of matrices, Linear and Multilinear Algebra 23 (1988), 2746. MR 943767 (89f:15020)
 [3]
 M. Eidelheit, On isomorphisms of rings of linear operators, Studia Math. 9 (1940), 97105. MR 0003467 (2:224d)
 [4]
 P. R. Halmos, A Hilbert space problem book, 2nd ed., SpringerVerlag, New York, 1982. MR 675952 (84e:47001)
 [5]
 A. A. Jafarian and A. R. Sourour, Spectrumpreserving linear maps, J. Funct. Anal. 66 (1986), 255261. MR 832991 (87m:47011)
 [6]
 R. V. Kadison, Isometries of operator algebra, Ann. of Math. (2) 54 (1951), 325338. MR 0043392 (13:256a)
 [7]
 , A generalized Schwarz inequality and algebraic invariants for operator algebras, Ann. of Math. (2) 56 (1952), 494503. MR 0051442 (14:481c)
 [8]
 R. V. Kadison and J. R. Ringrose, Fundamentals of the theory of operator algebras. I, Academic Press, New York, 1983. MR 719020 (85j:46099)
 [9]
 V. J. Pellegrini, Numerical range preserving operators on a Banach algebra, Studia Math. 54 (1975), 143147. MR 0388104 (52:8941)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199512312937
PII:
S 00029939(1995)12312937
Article copyright:
© Copyright 1995
American Mathematical Society
