Fuglede-Putnam type commutativity theorems for EP operators
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Abstract
Fuglede-Putnam theorem is not true in general for EP operators on Hilbert spaces. We prove that under some
conditions the theorem holds good. If the adjoint operation is replaced by Moore-Penrose inverse in the theorem,
we get Fuglede-Putnam type theorem for EP operators – however proofs are totally different. Finally, interesting
results on EP operators have been proved using several versions of Fuglede-Putnam type theorems for EP
operators on Hilbert spaces.
Keywords:
Fuglede-Putnam theorem, Moore-Penrose inverse, EP operatorMathematics Subject Classification:
Mathematics- Pages: 709-714
- Date Published: 01-01-2021
- Vol. 9 No. 01 (2021): Malaya Journal of Matematik (MJM)
K. G. Brock. A note on commutativity of a linear operator and its Moore-Penrose inverse. Numer. Funct. Anal. Optim., 11(7-8), (1990), 673-678.
S. L. Campbell and C. D. Meyer. EP operators and generalized inverses. Canad. Math. Bull, 18(3), (1975), $327-333$.
S. L. Campbell and C. D. Meyer. Generalized inverses of linear transformations, volume 56 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, (2009).
D. S. Djordjević. Products of EP operators on Hilbert spaces. Proc. Amer. Math. Soc., 129(6), (2001), 17271731.
D. S. Djordjević and J. J. Koliha. Characterizing Hermitian, normal and EP operators. Filomat, 21(1), (2007), $39-54$.
B. P. Duggal. A remark on generalised Putnam-Fuglede theorems. Proc. Amer. Math. Soc., 129(1), (2001), 83-87.
B. Fuglede. A commutativity theorem for normal operators. Proc. Nat. Acad. Sci. U. S. A., 36, (1950), 35-40.
W. B. Gong. A simple proof of an extension of the Fuglede-Putnam theorem. Proc. Amer. Math. Soc., 100(3), (1987), 599-600.
B. C. Gupta and S. M. Patel. On extensions of FugledePutnam theorem. Indian J. Pure Appl. Math., 19(1), (1988), 55-58.
R. E. Hartwig and I. J. Katz. On products of EP matrices. Linear Algebra Appl., 252, (1997), 339-345.
I. J. Katz and M. H. Pearl. On EPr and normal EPr matrices. J. Res. Nat. Bur. Standards Sect. B, 70B, (1966), $47-77$
$S$. Mecheri. An extension of the Fuglede-Putnam theorem to p-hyponormal operators. J. Pure Math., 21, (2004), $25-30$.
C. D. Meyer, Jr. Some remarks on $mathrm{EP}_r$ matrices, and generalized inverses. Linear Algebra and Appl., 3, (1970), $275-278$.
D. Mosić. Reflexive-EP elements in rings. Bull. Malays. Math. Sci. Soc., 40(2), (2017), 655-664.
M. H. Pearl. On generalized inverses of matrices. Proc. Cambridge Philos. Soc., 62, (1966), 673-677.
C. R. Putnam. On normal operators in Hilbert space. Amer. J. Math., 73, (1951), 357-362.
P. Sam Johnson and C. Ganesa Moorthy. Composition of operators with closed range. J. Anal., 14, (2006), 79-80.
P. Sam Johnson and A. Vinoth. Product and factorization of hypo-EP operators. Spec. Matrices, 6, (2018), 376382.
H. Schwerdtfeger. Introduction to Linear Algebra and the Theory of Matrices. P. Noordhoff, Groningen, (1950).
S. Xu, J. Chen, and J. Benítez. EP elements in rings with involution. Bull. Malays. Math. Sci. Soc., 42(6), (2019), 3409-3426.
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