Commutativity of compact selfadjoint operators

G. Greiner; W. Ricker

Studia Mathematica (1995)

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

Abstract

top
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 .

How to cite

top

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

top
  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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.