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Keywords:
hyponormal; totally $\ast $-paranormal; hypercyclic; operators
Summary:
In this paper we study some properties of a totally $\ast $-paranormal operator (defined below) on Hilbert space. In particular, we characterize a totally $\ast $-paranormal operator. Also we show that Weyl’s theorem and the spectral mapping theorem hold for totally $\ast $-paranormal operators through the local spectral theory. Finally, we show that every totally $\ast $-paranormal operator satisfies an analogue of the single valued extension property for $W^{2}(D,H)$ and some of totally $\ast $-paranormal operators have scalar extensions.
References:
[1] C. Apostol, L. A. Fialkow, D. A. Herrero and D. Voiculescu: Approximation of Hilbert space operators, Volume II. Research Notes in Mathematics 102, Pitman, Boston, 1984. MR 0735080
[2] S. C. Arora and J. K. Thukral: On a class of operators. Glasnik Math. 21 (1986), 381–386. MR 0896819
[3] S. K. Berberian: An extension of Weyl’s theorem to a class of not necessarily normal operators. Michigan Math J. 16 (1969), 273–279. DOI 10.1307/mmj/1029000272 | MR 0250094 | Zbl 0175.13603
[4] S. K. Berberian: The Weyl’s spectrum of an operator. Indiana Univ. Math. J. 20 (1970), 529–544. DOI 10.1512/iumj.1971.20.20044 | MR 0279623
[5] S. W. Brown: Hyponormal operators with thick spectrum have invariant subspaces. Ann. of Math. 125 (1987), 93–103. DOI 10.2307/1971289 | MR 0873378
[6] L. A. Coburn: Weyl’s theorem for non-normal operators. Michigan Math. J. 13 (1966), 285–288. DOI 10.1307/mmj/1031732778 | MR 0201969
[7] I. Colojoara and C. Foias: Theory of generalized spectral operators. Gordon and Breach, New York, 1968. MR 0394282
[8] J. B. Conway: Subnormal operators. Pitman, London, 1981. MR 0634507 | Zbl 0474.47013
[9] S. Djordjevic, I. Jeon and E. Ko: Weyl’s theorem through local spectral theory. Glasgow Math. J. 44 (2002), 323–327. MR 1902409
[10] B. P. Duggal: On the spectrum of $p$-hyponormal operators. Acta Sci. Math. (Szeged) 63 (1997), 623–637. MR 1480502 | Zbl 0893.47013
[11] J. Eschmeier: Invariant subspaces for subscalar operators. Arch. Math. 52 (1989), 562–570. DOI 10.1007/BF01237569 | MR 1007631 | Zbl 0651.47002
[12] P. R. Halmos: A Hilbert space problem book. Springer-Verlag, 1982. MR 0675952 | Zbl 0496.47001
[13] R. E. Harte: Invertibility and singularity. Dekker, New York, 1988. Zbl 0678.47001
[14] C. Kitai: Invariant closed sets for linear operators. Ph.D. Thesis, Univ. of Toronto, 1982.
[15] E. Ko: Algebraic and triangular $n$-hyponormal operators. Proc. Amer. Math. Soc. 123 (1995), 3473–3481. MR 1291779 | Zbl 0877.47015
[16] K. B. Laursen: Operators with finite ascent. Pacific J. Math. 152 (1992), 323–336. DOI 10.2140/pjm.1992.152.323 | MR 1141799 | Zbl 0783.47028
[17] K. B. Laursen: Essential spectra through local spectral theory. Proc. Amer. Math. Soc. 125 (1997), 1425–1434. DOI 10.1090/S0002-9939-97-03852-5 | MR 1389525 | Zbl 0871.47003
[18] K. K. Oberai: On the Weyl spectrum. Illinois J. Math. 18 (1974), 208–212. MR 0333762 | Zbl 0277.47002
[19] K. K. Oberai: On the Weyl spectrum (II). Illinois J. Math. 21 (1977), 84–90. MR 0428073 | Zbl 0358.47004
[20] M. Putinar: Hyponormal operators are subscalar. J. Operator Th. 12 (1984), 385–395. MR 0757441 | Zbl 0573.47016
[21] R. Lange: Biquasitriangularity and spectral continuity. Glasgow Math. J. 26 (1985), 177–180. DOI 10.1017/S0017089500005966 | MR 0798746 | Zbl 0583.47006
[22] B. L. Wadhwa: Spectral, $M$-hyponormal and decomposable operators. Ph.D. thesis, Indiana Univ., 1971.
[23] D. Xia: Spectral theory of hyponormal operators. Operator Theory 10, Birkhäuser-Verlag, 1983. MR 0806959 | Zbl 0523.47012
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