Maximal numerical range and quadratic elements in a \(C^\ast\)-algebra. (English) Zbl 1507.47011
Summary: In this paper, we give a description of the maximal numerical range of a hyponormal element and a characterization of a normaloid element in a \(C^\ast\)-algebra. We also give an explicit formula for the maximal numerical range of a quadratic operator acting on a complex Hilbert space. As a consequence, we determine the maximal numerical range of a rank-one operator.
MSC:
| 47A12 | Numerical range, numerical radius |
| 47B20 | Subnormal operators, hyponormal operators, etc. |
| 46L05 | General theory of \(C^*\)-algebras |
References:
| [1] | A. ABU-OMAR ANDP. Y. WU,Scalar approximants of quadratic operators with applications, Oper. Matrices,12(1) (2018), 253-262. · Zbl 1462.47001 |
| [2] | B. AUPETIT,A primer on spectral theory, New York (NY): Springer Verlag (1991). · Zbl 0715.46023 |
| [3] | A. BAGHDAD ANDM. C. KAADOUD,On the maximal numerical range of a hyponormal operator, Oper. Matrices,13(4) (2019), 1163-1171. · Zbl 1435.47010 |
| [4] | F. F. BONSALL ANDJ. DUNCAN,Numerical ranges of operators on normed spaces and of elements of normed algebras, London-New York: Cambridge University Press, (1971), (London mathematical society lecture note series; 2). · Zbl 0207.44802 |
| [5] | F. F. BONSALL ANDJ. DUNCAN,Numerical ranges II, New York-London: Cambridge University Press; (1973), (London mathematical society lecture notes series; 10). · Zbl 0262.47001 |
| [6] | J. T. CHAN ANDK. CHAN,An observation about normaloid operators, Oper. Matrices,11(3) (2017), 885-890. · Zbl 06787265 |
| [7] | C. K. FONG,On the essential maximal numerical range, Acta Sci. Math.,41(1979), 307-315. · Zbl 0444.47004 |
| [8] | K. E. GUSTAFSON ANDD. K. M. RAO,Numerical range: thefield of values of linear operators and matrices, Springer, New York, Inc: 1997. |
| [9] | P. R. HALMOS,Hilbert space problem book, New York: Van Nostrand (1967). · Zbl 0144.38704 |
| [10] | A. N. HAMED ANDI. M. SPITKOVSKY,On the maximal numerical range of some matrices, Electron J. Linear Algebra,34(2018), 288-303. · Zbl 1398.15021 |
| [11] | G. JI, N. LIU ANDZ. E. LI,Essential numerical range and maximal numerical range of the Aluthge transform, Linear Multilinear Algebra,55(4) (2007), 315-322. · Zbl 1119.47007 |
| [12] | D. N. KINGANGI,On Norm of Elementary Operator: An Application of Stampfli’s Maximal Numerical Range, Pure Appl. Math. J.,7(1) (2018), 6-10. |
| [13] | C. S. KUBRUSLY,Spectral Theory of Bounded Linear Operators, Birkh¨auser (2020). · Zbl 1454.47001 |
| [14] | J. ROOIN, S. KARAMI ANDM. G. AGHIDEH,A new approach to numerical radius of quadratic operators, Ann. Funct. Anal.,11(2020), 879-896. · Zbl 1498.47020 |
| [15] | I. M. SPITKOVSKY,A note on the maximal numerical range, Oper. Matrices,13(3) (2019), 601-605. · Zbl 1435.47012 |
| [16] | J. G. STAMPFLI ANDJ. P. WILLIAMS,Growth condition and the numerical range in Banach algebra, Tohoku Math. J.20(1968), 417-424. · Zbl 0175.43902 |
| [17] | J. G. STAMPFLI,The norm of derivation, Pacific J. Math.,33(1970), 737-747. · Zbl 0197.10501 |
| [18] | S. H. TSO ANDP. Y. WU,Matricial ranges of quadratic operators, Rocky Mountain J. Math.,29 (1999), 1139-1152 · Zbl 0957.47005 |
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