×

Lacunas in the support of the Weyl calculus for two Hermitian matrices. (English) Zbl 1059.47016

Let \(A_1\) and \(A_2\) be Hermitian matrices and \(A= A_1+ iA_2\). It is shown that \(\lambda\) belongs to the support of the Weyl functional calculus associated with \((A_1,A_2)\) if and only if for every neighbourhood \(U\) of \(\lambda\) there exist \(\zeta\in U\) and \(z\in G\) such that \(| z|\neq 1\) and \[ \text{det}((\zeta I- A)^* z+(\zeta I- A))= 0. \] It follows that the spectrum of \(A\) is contained in the support of the Weyl functional calculus with equality if and only if \(A\) is normal. Examples are exhibited where this support has gaps. In order to achieve the mentioned results, the connection between Clifford analysis and the Weyl functional calculus for a \(d\)-tuple of bounded selfajoint operators is used.

MSC:

47A60 Functional calculus for linear operators
46H30 Functional calculus in topological algebras
47A25 Spectral sets of linear operators
30G35 Functions of hypercomplex variables and generalized variables
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Joswig, J. Austral. Math. Soc. 65 pp 267– (1998)
[2] DOI: 10.1002/mana.19510060306 · Zbl 0044.16201 · doi:10.1002/mana.19510060306
[3] Brackx, Clifford analysis (1982)
[4] Bochnak, Géométrie algébrique réelle (1987)
[5] DOI: 10.1002/cpa.3160220302 · Zbl 0167.10003 · doi:10.1002/cpa.3160220302
[6] Bazer, Comm. Pure Appl. Math. 20 pp 329– (1967)
[7] Baumgärtel, Analytic perturbation theory for matrices and operators (1985) · Zbl 0591.47013
[8] DOI: 10.1063/1.1706202 · Zbl 0121.44601 · doi:10.1063/1.1706202
[9] DOI: 10.1007/BF02392039 · Zbl 0266.35045 · doi:10.1007/BF02392039
[10] Vassiliev, Ramified integrals, singularities and lacunas (1995) · Zbl 0935.32026 · doi:10.1007/978-94-011-0213-1
[11] DOI: 10.1007/BF02394570 · Zbl 0191.11203 · doi:10.1007/BF02394570
[12] DOI: 10.2307/2036144 · Zbl 0164.16604 · doi:10.2307/2036144
[13] DOI: 10.1016/0022-1236(69)90013-5 · Zbl 0191.13403 · doi:10.1016/0022-1236(69)90013-5
[14] Sommen, Ann. Polon. Math. 49 pp 101– (1988)
[15] Ahlfors, Complex analysis (1966)
[16] Shafarevich, Basic algebraic geometry (1977) · Zbl 0362.14001
[17] Kato, Perturbation theory for linear operators (1980) · Zbl 0435.47001
[18] DOI: 10.1112/plms/s3-64.1.70 · Zbl 0703.30043 · doi:10.1112/plms/s3-64.1.70
[19] DOI: 10.1080/03081089308818247 · Zbl 0796.15015 · doi:10.1080/03081089308818247
[20] Jefferies, Studia Math. 136 pp 99– (1999)
[21] Jefferies, Bull. Austral. Math. Soc. 57 pp 329– (1998)
[22] DOI: 10.4064/sm144-3-1 · Zbl 0985.47014 · doi:10.4064/sm144-3-1
[23] DOI: 10.1090/S0002-9939-96-03143-7 · Zbl 0848.47014 · doi:10.1090/S0002-9939-96-03143-7
[24] Hillman, Linear Algebra Appl. 267 pp 317– (1997) · Zbl 0885.15017 · doi:10.1016/S0024-3795(97)80055-9
[25] Gustafson, Numerical range, the field of values of linear operators and matrices (1997)
[26] Greiner, Studia Math. 112 pp 109– (1995)
[27] Dunford, Linear operators. Part II. (1963)
[28] Bremermann, Distributions, complex variables and Fourier transforms (1964)
[29] DOI: 10.1007/BF01378781 · Zbl 0847.47009 · doi:10.1007/BF01378781
[30] DOI: 10.1007/BF01571652 · Zbl 0016.06201 · doi:10.1007/BF01571652
[31] Petrovsky, Mat. Sbornik 17 pp 289– (1945)
[32] Nelson, Functional analysis and related fields, Proceedings of a conference in honour of Professor Marshal Stone, Univ. of Chicago, May 1968 pp 172– (1970)
[33] DOI: 10.1073/pnas.18.3.246 · Zbl 0004.05003 · doi:10.1073/pnas.18.3.246
[34] Li, Rev. Mat. Iberoamericana 10 pp 665– (1994) · Zbl 0817.42008 · doi:10.4171/RMI/164
[35] John, Plane waves and spherical means applied to partial differential equations (1955) · Zbl 0067.32101
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.