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1994-1995 | 112 | 2 | 109-125
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Commutativity of compact selfadjoint operators

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The relationship between the joint spectrum γ(A) of an n-tuple $A = (A_1,..., A_n)$ of selfadjoint operators and the support of the corresponding Weyl calculus T(A) : f ↦ f(A) is discussed. It is shown that one always has γ(A) ⊂ supp (T(A)). Moreover, when the operators are compact, equality occurs if and only if the operators $A_j$ mutually commute. In the non-commuting case the equality fails badly: While γ(A) is countable, supp(T(A)) has to be an uncountable set. An example is given showing that, for non-compact operators, coincidence of γ(A) and supp (T(A)) no longer implies commutativity of the set ${A_i}$ .
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Bibliografia
  • [1] R. F. V. Anderson, The Weyl functional calculus, J. Funct. Anal. 4 (1969), 240-267.
  • [2] G. B. Folland, Harmonic Analysis in Phase Space, Ann. of Math. Stud. 122, Princeton Univ. Press, Princeton, 1989.
  • [3] G. Greiner and W. J. Ricker, Joint spectral sets and commutativity of systems of (2× 2) selfadjoint matrices, Linear and Multilinear Algebra 36 (1993), 47-58.
  • [4] B. R. F. Jefferies and W. J. Ricker, Commutativity for systems of (2×2) selfadjoint matrices, ibid. 35 (1993), 107-114.
  • [5] A. McIntosh and A. J. Pryde, A functional calculus for several commuting operators, Indiana Univ. Math. J. 36 (1987), 421-439.
  • [6] A. McIntosh, A. J. Pryde and W. J. Ricker, Comparison of joint spectra for certain classes of commuting operators, Studia Math. 88 (1988), 23-36.
  • [7] A. McIntosh, A. J. Pryde and W. J. Ricker, Systems of operator equations and perturbation of spectral subspaces of commuting operators, Michigan Math. J. 35 (1988), 43-65.
  • [8] A. J. Pryde, A non-commutative joint spectral theory, Proc. Centre Math. Anal. (Canberra) 20 (1988), 153-161.
  • [9] A. J. Pryde, Inequalities for exponentials in Banach algebras, Studia Math. 100 (1991), 87-94.
  • [10] W. J. Ricker, The Weyl calculus and commutativity for systems of selfadjoint matrices, Arch. Math. (Basel) 61 (1993), 173-176.
  • [11] H. H. Schaefer, Banach Lattices and Positive Operators, Springer, 1974.
  • [12] M. E. Taylor, Functions of several selfadjoint operators, Proc. Amer. Math. Soc. 19 (1968), 91-98.
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bwmeta1.element.bwnjournal-article-smv112i2p109bwm
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