On mappings preserving orthogonality of non-singular vectors

Alpers, Burkhard; Schröder, Eberhard M.

doi:10.1007/bf01258504

Cited by 6 publications

(4 citation statements)

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“…If X is complex and has finite dimension n, then operators on X can be identified a s n x n complex matrices. Molnar showed that if n > 3, bijective zero product preservers on the n x n rank one idempotent matrices have the form (2.1) (ay) H+ S(o(a u ))S- 1 for some invertible matrix S and field automorphism a.…”

Section: >(A)(j>(b)mentioning

confidence: 99%

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Mappings on matrices: invariance of functional values of matrix products

Chan

Li²,

Sze³

2006

J. Aust. Math. Soc.

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Let M n be the algebra of all n x n matrices over a field IF, where n > 2. Let : S -*• M n such that F((A)(j>(B)) = F(AB)for various families of functions F including all the unitary similarity invariant functions on real or complex matrices. Very often, these mappings have the form A i-> iJ.(A)S(a(ajj))S~{ for all A = (a,-y) e S for some invertible S e M n , field monomorphism a of F, and an P-valued mapping \x defined on S. For real matrices, a is often the identity map; for complex matrices, a is often the identity map or the conjugation map: z i-> z. A key idea in our study is reducing the problem to the special case when F : M n ->• {0, 1} is defined by F(X) = 0, if X = 0, and F(X) = 1 otherwise. In such a case, one needs to characterize : S -> M n such that 4>(A)4>(B) = 0 if and only if AS = 0. We show that such a map has the standard form described above on rank one matrices in S.2000 Mathematics subject classification: primary 15A04, 15A60, 15A18.

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Section: >(A)(j>(b)mentioning
confidence: 99%

~~“…Next we show that our main theorem can be used to study mappings on Ppreserving orthogonality; see [1]. We write u = v if u is a scalar multiple of v.…”~~

Section: Proof By Lemma 23 We See That (P(xy') = /J(xy')f(x)g(y)'mentioning
confidence: 99%

Mappings on matrices: invariance of functional values of matrix products
Chan
¹
,
Li²,
Sze³
2006
J. Aust. Math. Soc.
45
1
17
0
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~~“…We now prove a theorem which even holds true in certain non-real situations as was shown, based on other methods, by A. Alpers and E. M. Schröder in [2]. Theorem 2.…”~~

Section: Isometriesmentioning
confidence: 97%

~~“…We define the elliptic distance ε (x, y) of x, y ∈ X 0 := X\{0} by means of ε (x, y) ∈ 0, π 2 and cos ε (x, y) = |xy| x · y (2) with x := √ x 2 .…”~~

Section: Introductionmentioning
confidence: 99%

Elliptic distances in Hilbert spaces
Benz¹
2000
Aequ. math.
3
0
1
0
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On Wigner's theorem: Remarks, complements, comments, and corollaries
Rätz
¹

1996
Aeq. Math.
43
0
27
0
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In this paper we present a unified treatment of Wigner's unitarity-antiunitarity theorem simultaneously in the real and the complex case. Its elementary nature, emphasized by V. Bargmann in 1964, is underlined here by removing unnecessary hypotheses, the most important being the completeness of the inner product spaces involved. At the end, we shall obtain connections to some recent results in geometry. 2 J. RATZ AEQ. MATH.

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On mappings preserving orthogonality of non-singular vectors

Cited by 6 publications

References 6 publications

Mappings on matrices: invariance of functional values of matrix products

Mappings on matrices: invariance of functional values of matrix products

Elliptic distances in Hilbert spaces

On Wigner's theorem: Remarks, complements, comments, and corollaries

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