# Commutativity of compact selfadjoint operators

Studia Mathematica (1995)

• Volume: 112, Issue: 2, page 109-125
• ISSN: 0039-3223

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## Abstract

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The relationship between the joint spectrum γ(A) of an n-tuple $A=\left({A}_{1},...,{A}_{n}\right)$ 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 ${A}_{i}$ .

## How to cite

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Greiner, G., and Ricker, W.. "Commutativity of compact selfadjoint operators." Studia Mathematica 112.2 (1995): 109-125. <http://eudml.org/doc/216141>.

@article{Greiner1995,
abstract = {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 $\{A_i\}$ .},
author = {Greiner, G., Ricker, W.},
journal = {Studia Mathematica},
keywords = {joint spectrum; Weyl calculus; non-compact operators},
language = {eng},
number = {2},
pages = {109-125},
title = {Commutativity of compact selfadjoint operators},
url = {http://eudml.org/doc/216141},
volume = {112},
year = {1995},
}

TY - JOUR
AU - Greiner, G.
AU - Ricker, W.
TI - Commutativity of compact selfadjoint operators
JO - Studia Mathematica
PY - 1995
VL - 112
IS - 2
SP - 109
EP - 125
AB - 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 ${A_i}$ .
LA - eng
KW - joint spectrum; Weyl calculus; non-compact operators
UR - http://eudml.org/doc/216141
ER -

## References

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1. [1] R. F. V. Anderson, The Weyl functional calculus, J. Funct. Anal. 4 (1969), 240-267. Zbl0191.13403
2. [2] G. B. Folland, Harmonic Analysis in Phase Space, Ann. of Math. Stud. 122, Princeton Univ. Press, Princeton, 1989. Zbl0682.43001
3. [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. Zbl0794.15007
4. [4] B. R. F. Jefferies and W. J. Ricker, Commutativity for systems of (2×2) selfadjoint matrices, ibid. 35 (1993), 107-114. Zbl0796.15015
5. [5] A. McIntosh and A. J. Pryde, A functional calculus for several commuting operators, Indiana Univ. Math. J. 36 (1987), 421-439. Zbl0694.47015
6. [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. Zbl0665.47002
7. [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. Zbl0657.47021
8. [8] A. J. Pryde, A non-commutative joint spectral theory, Proc. Centre Math. Anal. (Canberra) 20 (1988), 153-161. Zbl0705.47013
9. [9] A. J. Pryde, Inequalities for exponentials in Banach algebras, Studia Math. 100 (1991), 87-94. Zbl0787.47005
10. [10] W. J. Ricker, The Weyl calculus and commutativity for systems of selfadjoint matrices, Arch. Math. (Basel) 61 (1993), 173-176. Zbl0819.47005
11. [11] H. H. Schaefer, Banach Lattices and Positive Operators, Springer, 1974. Zbl0296.47023
12. [12] M. E. Taylor, Functions of several selfadjoint operators, Proc. Amer. Math. Soc. 19 (1968), 91-98. Zbl0164.16604

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