[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.5666/KMJ.2017.57.3.457

On Normal Products of Selfadjoint Operators

Jung, Il Bong (Department of Mathematics, Kyungpook National University)
Mortad, Mohammed Hichem (Department of Mathematics, University of Oran 1 (Ahmed Ben Bella))
Stochel, Jan (Instytut Matematyki, Uniwersytet Jagiellonski)

Publication Information

Kyungpook Mathematical Journal / v.57, no.3, 2017 , pp. 457-471 More about this Journal

Abstract

A necessary and sufficient condition for the product AB of a selfadjoint operator A and a bounded selfadjoint operator B to be normal is given. Various properties of the factors of the unitary polar decompositions of A and B are obtained in the case when the product AB is normal. A block operator model for pairs (A, B) of selfadjoint operators such that B is bounded and AB is normal is established. The case when both operators A and B are bounded is discussed. In addition, the example due to Rehder is reexamined from this point of view.

Keywords

Selfadjoint operator; normal operator; unitary polar decomposition; block operator model; normal product;

Citations & Related Records

Reference

1	M. H. Mortad, On some product of two unbounded self-adjoint operators, Integr. Equ. Oper. Theory, 64(2009), 399-408. DOI
2	C. R. Putnam, On normal operators in Hilbert space, Amer. J. Math., 73(1951), 357-362. DOI
3	W. Rehder, On the Product of Self-adjoint Operators, Internat. J. Math. Math. Sci., 5(1982), 813-816. DOI
4	M. H. Mortad, An application of the Putnam-Fuglede theorem to normal products of selfadjoint operators, Proc. Amer. Math. Soc., 131(2003), 3135-3141. DOI
5	E. Albrecht, P. G. Spain, When products of self-adjoints are normal, Proc. Amer. Math. Soc., 128(2000), 2509-2511. DOI
6	M. Sh. Birman, M. Z. Solomjak, Spectral theory of selfadjoint operators in Hilbert space, D. Reidel Publishing Co., Dordrecht, 1987.
7	J. Bognar, Indefinite inner product spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 78. Springer-Verlag, New York-Heidelberg, 1974.
8	S. Dehimi, M. H. Mortad, Right (or left) invertibility of bounded and unbounded operators and applications to the spectrum of products, Complex Anal. Oper. Theory (2017). https://doi.org/10.1007/s11785-017-0687-z. DOI
9	T. Furuta, Invitation to linear operators, Taylor & Francis, Ltd., London, 2001.
10	K. Gustafson, M. H. Mortad, Conditions implying commutativity of unbounded self-adjoint operators and related topics, J. Operator Theory, 76(2016), 159-169. DOI
11	Z. J. Jab lonski, I. B. Jung, J. Stochel, Unbounded quasinormal operators revisited, Integr. Equ. Oper. Theory, 79(2014), 135-149. DOI