Fuglede – Putnam type theorems for extension of $M$-hyponormal operators

S. Mecheri; T. Prasad

doi:10.37863/umzh.v74i1.2355

S. Mecheri Mohamed El Bachir El Ibrahimi Univ., Bordj Bou Arreridj, Algeria
T. Prasad Univ. Calicut, Kerala, India https://orcid.org/0000-0002-9237-6018

DOI: https://doi.org/10.37863/umzh.v74i1.2355

Анотація

УДК 517.9

Теореми типу Фугледе–Путнама для розширень $M$ -гiпонормальних операторiв

Розглянуто $k$-квазі-$M$-гіпонормальний оператор $T \in B(\mathcal{H})$ такий, що $TX = XS$ для деякого $X \in B(\mathcal{K},\mathcal{H}),$ та доведено теорему типу Фугледе–Путнама, коли спряженим до $S \in B(\mathcal{K})$ є або $k$-квазі-$M$-гіпонормальний, або домінуючий оператор.
Також показано, що два квазіподібні $k$-квазі-$M$-гіпонормальні оператори мають однакові суттєві спектри.

Посилання

R. G. Douglas, On majorization, factorization, and range inclusion of operators on Hilbert space, Proc. Amer. Math. Soc., 17, 413 – 415 (1966), https://doi.org/10.2307/2035178 DOI: https://doi.org/10.1090/S0002-9939-1966-0203464-1

B. C. Gupta, An extension of Fuglede – Putnam theorem and normality of operators, Indian J. Pure. and Appl. Math., 14, № 11, 1343 – 1347 (1983).

B. C. Gupta, P. B. Ramanujan, On $k$-quasihyponormal operators-II, Bull. Aust. Math. Soc., 83, 514 – 516 (1981), https://doi.org/10.1017/S0004972700007437 DOI: https://doi.org/10.1017/S0004972700007437

P. R. Halmos, A Hilbert space problem book, second ed., Springer-Verlag, New York (1982). DOI: https://doi.org/10.1007/978-1-4684-9330-6

S. Mecheri, On $k$-quasi-$M$-hyponormal operators, Math. Inequal. Appl., 16, 895 – 902 (2013), https://doi.org/10.7153/mia-16-69 DOI: https://doi.org/10.7153/mia-16-69

S. Mecheri, Fuglede – Putnams theorem for class $A$ operators, Colloq. Math., 138, 183 – 191 (2015), https://doi.org/10.4064/cm138-2-3 DOI: https://doi.org/10.4064/cm138-2-3

A. H. Kim, I. H. Kim, Essential spectra of quasisimilar $(p, k)$ quasihyponormal operators, J. Inequality and Appl., 1 – 7 (2006), https://doi.org/10.1155/JIA/2006/72641 DOI: https://doi.org/10.1155/JIA/2006/72641

K. B Laursen, M. M. Neumann, An introduction to local spectral theory, Clarendon Press, Oxford (2000).

M. S. Moslehian, S. M. S. Nabavi, Sales, Fuglede – Putnam type theorems via the Aluthge transform, Positivity, 17, № 1, 151 – 162 (2013), https://doi.org/10.1007/s11117-011-0154-4 DOI: https://doi.org/10.1007/s11117-011-0154-4

R. L. Moore, D. D. Rogers, T. T. Trent, A note on intertwining M-hyponormal operators, Proc. Amer. Math. Soc., 83, 514 – 516 (1981), https://doi.org/10.2307/2044108 DOI: https://doi.org/10.2307/2044108

C. R. Putnam, Ranges of normal and subnormal operators, Michigan Math. J., 18, 33 – 36 (1971). DOI: https://doi.org/10.1307/mmj/1029000584

C. R. Putnam, Hyponormal contraction and strong power convergence, Pacif. J. Math., 57, 531 – 538 (1975). DOI: https://doi.org/10.2140/pjm.1975.57.531

M. Radjabalipour, Ranges of hyponormal operators, Illinois J. Math., 21, 70 – 75 (1977). DOI: https://doi.org/10.1215/ijm/1256049502

M. Radjabalipour, On majorization and normality of operators, Proc. Amer. Math. Soc., 62, 105 – 110 (1977), https://doi.org/10.2307/2041955 DOI: https://doi.org/10.1090/S0002-9939-1977-0430851-6

K. Takahashi, On the converse of Fuglede – Putnam theorem, Acta Sci. Math. (Szeged), 43, 123 – 125 (1981).

J. G. Stampfli, B. L. Wadhwa, On dominant operators, Monatsh. Math., 84, 143 – 153 (1977), https://doi.org/10.1007/BF01579599 DOI: https://doi.org/10.1007/BF01579599

J. G. Stampfli, B. L. Wadhwa, An asymmetric Putnam – Fuglede theorem for dominant operators, Indiana Univ. Math. J., 25, 359 – 365 (1976), https://doi.org/10.1512/iumj.1976.25.25031 DOI: https://doi.org/10.1512/iumj.1976.25.25031

S. Jo, Y. Kim, E. Ko, On Fuglede – Putnam properties, Positivity, 19, 911 – 925 (2015), https://doi.org/10.1007/s11117-015-0335-7 DOI: https://doi.org/10.1007/s11117-015-0335-7

K. Tanahashi, S. M. Patel, A. Uchiyama, On extensions of some Fuglede – Putnam type theorems involving $(p, k)$-quasihyponormal, spectral, and dominant operators, Math. Nachr., 282, № 7, 1022 – 1032 (2009), https://doi.org/10.1002/mana.200610787 DOI: https://doi.org/10.1002/mana.200610787

L. M. Yang, Quasisimilarity of hyponormal and subdecomposable operators, J. Funct. Anal., 112, 204 – 217 (1993), https://doi.org/10.1006/jfan.1993.1030 DOI: https://doi.org/10.1006/jfan.1993.1030

T. Yoshino, Remark on the generalized Putnam – Fuglede theorem, Proc. Amer. Math. Soc., 95, 571 – 572 (1985), https://doi.org/10.2307/2045845 DOI: https://doi.org/10.1090/S0002-9939-1985-0810165-7

F. Zuo, S. Mecheri, Spectral Properties of $k$-quasi-$M$-hyponormal operators, Complex Anal. and Oper. Theory, 12, 1877 – 1887 (2018), https://doi.org/10.1007/s11785-017-0686-0 DOI: https://doi.org/10.1007/s11785-017-0686-0