Linear maps on the space of all bounded observables preserving maximal deviation

Molnár, Lajos; Barczy, Matyas

doi:10.1016/s0022-1236(03)00213-1

Cited by 12 publications

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“…To see this property, observe that diam σ(A) = diam σ(A + αI) for any A ∈ S(H) and α ∈ R. Now using (6), the linearity of Φ and the equality Φ(I) = I, it follows that Φ preserves the diameter of the spectrum. The result [12,Theorem 2] describes the structure of those linear bijections on the space of all self-adjoint operators on a Hilbert space which leave the so-called maximal deviation invariant. According to [12,Lemma 1], this quantity is the half of the diameter of the spectrum, therefore the former statement tells us the general form of those linear automorphisms of S(H) which preserve the quantity diam σ(A) (A ∈ S(H)).…”

Section: Elementary Considerations Show That φ L Is a Surjective Isommentioning

confidence: 99%

“…The result [12,Theorem 2] describes the structure of those linear bijections on the space of all self-adjoint operators on a Hilbert space which leave the so-called maximal deviation invariant. According to [12,Lemma 1], this quantity is the half of the diameter of the spectrum, therefore the former statement tells us the general form of those linear automorphisms of S(H) which preserve the quantity diam σ(A) (A ∈ S(H)). Applying [12, Theorem 2] to Φ we obtain that there exist a unitary or an antiunitary operator U on H, a linear functional Λ : S(H) → R and a number τ ∈ {−1, 1} such that Φ can be written in the form Φ(A) = τ U AU * + Λ(A)I (A ∈ S(H)).…”

Section: Elementary Considerations Show That φ L Is a Surjective Isommentioning

confidence: 99%

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Isometries of the spaces of self-adjoint traceless operators

Nagy

2015

Linear Algebra and its Applications

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In this paper we describe the structure of all isometries of the space of self-adjoint traceless operators on a finite dimensional Hilbert space under the metrics coming from the operator norm or the Schatten norms. We prove that up to a translation and multiplication by -1, they are unitary or antiunitary similarity transformations.Keywords: Isometries, self-adjoint traceless operators, operator norm, Schatten norms 2010 MSC: Primary: 15A86, Secondary: 47B49.Throughout this paper H denotes a finite dimensional complex Hilbert space. Recently, in [7] Hatori has determined the structure of isometries of the special unitary group SU(H) on H. This group -which is the set of unitary operators on H with determinant 1 -is strongly connected to the linear space S 0 (H) of all self-adjoint operators on H whose trace is 0. In fact, it is a folk result that for an operator W on H one has W ∈ SU(H) if and only if W = e iA , where A ∈ S 0 (H) is an element. So, roughly speaking we can say that S 0 (H) consists of the arguments of operators in SU(H). Moreover, it is also well-known that the Lie algebra of the Lie group SU(H) consists of the elements of S 0 (H) multiplied by i. In the light of these facts, it is not surprising that there is a connection between the isometries of S 0 (H) and of SU(H). Indeed, as [7, Lemma 2.1.] shows, any isometry of SU(H) with respect to the metric coming from the operator norm gives rise to a surjective linear isometry of S 0 (H). The proof of [7, Theorem 1.1.] on the structure of such maps of SU(H) is based on several lemmas and involves long computations which were necessary, since, as far as we are concerned there is no structural result for the isometries of S 0 (H) in the literature. Motivated mainly by the latter facts, in this paper we describe the general form of those transformations. Having a closer look at the mentioned proof and at Theorem below, it turns out that applying that result, the verification of [7, Theorem 1.1.] can be greatly shortened.The result in this paper is classified into the broad field of isometries of normed spaces which has a vast literature extending to several parts of mathematics. For a collection of results in the latter area the reader can consult, e.g. the books [5,6]. A particular subarea of this branch of mathematics is that of the isometries on spaces of bounded linear operators which is investigated by many authors. We remark that in the rest of the present paragraph by an isometry we mean not just an arbitrary distance preserving map, but a transformation leaving the metric induced by the operator norm invariant. The description of surjective linear isometries of classical normed spaces of operators is known. In cases of the algebra of bounded linear operators on H and the Email address: nagyg@science.unideb.hu (Gergő Nagy)

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Section: Elementary Considerations Show That φ L Is a Surjective Isommentioning

confidence: 99%

Section: Elementary Considerations Show That φ L Is a Surjective Isommentioning

confidence: 99%

Isometries of the spaces of self-adjoint traceless operators

Nagy

2015

Linear Algebra and its Applications

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“…Theorem 1.1 was first established for the structure of self-adjoint operators acting on a Hilbert space algebra B(H), where dim H ≥ 3 by Molnár and Barczy. 5,7 Later Molnár extended this result to all von Neumann algebras without type I 2 direct summand. The insight of these authors was vital for our arguments.…”

Section: Introductionmentioning

confidence: 90%

Linear maps preserving maximal deviation and the Jordan structure of quantum systems

Hamhalter

2012

Journal of Mathematical Physics

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Articles you may be interested inInfluence of dense quantum plasmas on fine-structure splitting of Lyman doublets of hydrogenic systems Phys. Plasmas 22, 054503 (2015); 10.1063/1.4921647 Efficient methods for including quantum effects in Monte Carlo calculations of large systems: Extension of the displaced points path integral method and other effective potential methods to calculate properties and distributions A generalized statistical complexity measure: Applications to quantum systems In the algebraic approach to quantum theory, a quantum observable is given by an element of a Jordan algebra and a state of the system is modelled by a normalized positive functional on the underlying algebra. Maximal deviation of a quantum observable is the largest statistical deviation one can obtain in a particular state of the system. The main result of the paper shows that each linear bijective transformation between JBW algebras preserving maximal deviations is formed by a Jordan isomorphism or a minus Jordan isomorphism perturbed by a linear functional multiple of an identity. It shows that only one numerical statistical characteristic has the power to determine the Jordan algebraic structure completely. As a consequence, we obtain that only very special maps can preserve the diameter of the spectra of elements. Nonlinear maps preserving the pseudometric given by maximal deviation are also described. The results generalize hitherto known theorems on preservers of maximal deviation in the case of self-adjoint parts of von Neumann algebras proved by Molnár. C 2012 American Institute of Physics. [http://dx.

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“…If we take supremum of the square roots of the variances of A over the set of all pure states, we obtain the so-called maximal deviation of A. Bijective linear maps on the space B s (H ) of all bounded observables preserving the maximal deviation were completely described in [5]. That result has recently been generalized significantly in [4] for the case of observables belonging to given von Neumann algebras.…”

Section: Proposition Letmentioning

confidence: 96%

Transformations on Bounded Observables Preserving Measure of Compatibility

Molnár

Timmermann

2011

Int J Theor Phys

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In this paper we describe the structure of all bijective nonlinear maps on the space of all bounded self-adjoint operators acting on a complex separable Hilbert space of dimension at least 3 which preserve a measure of commutativity, namely, the norm of the commutator of operators.Keywords Bounded observables · Self-adjoint operators · Compatibility · Commutativity · Preservers Statement of the ResultIn the Hilbert space formalism of the mathematical foundations of quantum mechanics, the most important roles are played by Hilbert spaces, their projections, self-adjoint operators, and density operators. In fact, to every quantum system there corresponds a complex Hilbert space H . The unit vectors or, in an equivalent formulation, the rank-one projections on H represent the pure states of the system and the (bounded) self-adjoint operators correspond to the (bounded) observable physical quantities. We also recall that the so-called density operators (positive operators with unit trace) represent the mixed states of the system.Transformations of quantum structures that preserve characteristic operations, relations, quantities, etc. relating to the elements of the structures can be viewed as symmetries of

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Linear maps on the space of all bounded observables preserving maximal deviation

Abstract: In this paper we determine all the bijective linear maps on the space of bounded observables which preserve a fixed moment or the variance. Nonlinear versions of the corresponding results are also presented.

Cited by 12 publications

References 20 publications

Isometries of the spaces of self-adjoint traceless operators

Isometries of the spaces of self-adjoint traceless operators

Linear maps preserving maximal deviation and the Jordan structure of quantum systems

Transformations on Bounded Observables Preserving Measure of Compatibility

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