Algebraic extension of $\mathcal{A}^{*}_{n}$ operator
DOI:
https://doi.org/10.5269/bspm.43345Abstract
$T\in L(H_{1}\oplus H_{2})$ is said to be an algebraic extension of a $\mathcal{A}^{*}_{n}$ operator if $$ T = \begin{pmatrix} T_{1} & T_{2} \\
O & T_{3} \end{pmatrix} $$ is an operator matrix on $H_{1}\oplus H_{2}$, where $T_{1}$ is a $\mathcal{A}^{*}_{n}$ operator and $T_{3}$ is a algebraic.
In this paper, we study basic and spectral properties of an algebraic extension of a $\mathcal{A}^{*}_{n}$ operator. We show that every algebraic extension of a $\mathcal{A}^{*}_{n}$ operator has SVEP, is polaroid and satisfies Weyl's theorem.
References
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2. Aiena, P.; Aponte, E.; Bazan, E., Weyl type theorems for left and right polaroid operators, Integral Equations Oper. Theory 66, 1-20, (2010). https://doi.org/10.1007/s00020-009-1738-2
3. Aiena, P.; PeËœna, P., Variations on Weyls theorem, J. Math. Anal. Appl. 324 (1), 566-579, (2006). https://doi.org/10.1016/j.jmaa.2005.11.027
4. Aiena, P., Fredholm and Local Spectral Theory with Applications to Multipliers, Kluwer, (2004).
5. Cao, X. H., Analytically Class A operators and Weyl's theorem, J. Math. Anal. Appl. 320, 795-803, (2006). https://doi.org/10.1016/j.jmaa.2005.07.056
6. Foias, C.; Frazho, A.E., The Commutant Lifting Approach to Interpolation Problem, Basel, Birkh¨auser Verlag, (1990). https://doi.org/10.1007/978-3-0348-7712-1
7. Hansen, F., An operator inequality, Math. Ann. 246, 249-250, (1980). https://doi.org/10.1007/BF01371046
8. Heuser, H., Functional Analysis, Marcel Dekker, New York, (1982).
9. Hoxha, I.; Braha, N. L., On k-Quasi Class A∗n Operators , Bulletin of Mathematical Analysis and Applications, Volume 6 Issue 1, Pages 23-33, (2014).
10. Han, J. K.; Lee, H. Y.; Lee, W. Y., Invertible completions of 2 × 2 upper triangular operator matrices Proceedings of the American Mathematical Society, vol. 128, no. 1, pp. 119-123, (2000). https://doi.org/10.1090/S0002-9939-99-04965-5
11. Lee, W. Y.; Lee, S. H., A spectral mapping theorem for the Weyl spectral, Glasgow Math. J. 38, no. 1, 61-64, (1996). https://doi.org/10.1017/S0017089500031268
12. Rashid, M. H. M., Property (ω) and quasi class (A, k) operators, Rev. Un. Mat. Argentina, Vol. 52, Nu 1, 133-142, (2011).
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2021-12-17
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