×

On the iterated mean transforms of operators. (English) Zbl 1512.47040

Let \(T=U|T|\) be the polar decomposition of an operator \(T\in \mathcal{L}(H)\), where \(U\) is assumed to be unitary. The main objective of the authors is to study properties of the \(k\)-iterated weighted mean transform \((\widehat{T}_{s,t}^{(k)})\) of \(T\), where \[\widehat{T}_{s,t}=sU|T|+t|T|U.\] In particular, they prove that the polar decomposition of the \(k\)-iterated weighted mean transform of \(T\) is given by \[ \widehat{T}_{s,t}^{(k)}=U\left[\sum_{j=0}^{k} C^j_k s^{k-j} t^j U^{*j}|T|U^j\right]. \]

MSC:

47B20 Subnormal operators, hyponormal operators, etc.
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] A. ALUTHGE,On p -hyponormal operators for0<p<1 , Inter. Equ. Oper. Theory13(1990), 307- 315. · Zbl 0718.47015
[2] S. K. BERBERIAN,Approximate proper vectors, Proc. Amer. Math. Soc.13(1962), 111-114. · Zbl 0166.40503
[3] I. COLOJOARA AND˘C. FOIAS¸,Theory of generalized spectral operators, Gordon and Breach, New York, 1968. · Zbl 0189.44201
[4] C. FOIAS¸, I. B. JUNG, E. KO ANDC. PEARCY,Complete contractivity of maps associated with the Aluthge and Duggal transforms, Pacific J. Math.209(2003), 249-259. · Zbl 1066.47037
[5] M. ITO, T. YAMAZAKI ANDM. YANAGIDA,On the polar decomposition of the Aluthge trasformation and related results, J. Operator Theory51(2004), 303-319. · Zbl 1104.47004
[6] I. B. JUNG, E. KO ANDC. PEARCY,Aluthge transforms of operators, Inter. Equ. Oper. Theory 37(2000), 449-456. · Zbl 0996.47008
[7] I. B. JUNG, E. KO ANDC. PEARCY,Spectral pictures of Aluthge transforms of operators, Inter. Equ. Oper. Theory40(2001), 52-60. · Zbl 1130.47300
[8] I. B. JUNG, E. KO ANDC. PEARCY,The iterated Aluthge transform of an operator, Inter. Equ. Oper. Theory45(2003), 375-387. · Zbl 1030.47021
[9] S. JUNG, E. KO ANDS. PARK,Subscalarity of operator transforms, Math. Nachr.288(2015), 2042- 2056. · Zbl 1357.47008
[10] E. KO ANDM. LEE,On backward Aluthge iterates of hyponormal operators, Math. Inequal. Appl. 18(2015), 1121-1133. · Zbl 1408.47006
[11] S. LEE, W. LEE ANDJ. YOON,The mean transform of bounded linear operators, J. Math. Anal. Appl.410(2014), 70-81. · Zbl 1327.47016
[12] K. B. LAURSEN ANDM. M. NEUMANN,Introduction to Local spectral theory, London Math. Soc. Monograghs New Series. Claredon Press, Oxford, 2000. · Zbl 0957.47004
[13] S. MATHEW ANDM. S. BALASUBRAMANI,On the polar decomposition of the Duggal transformation and related results, Oper. Matrices3(2009), 215-225. · Zbl 1200.47030
[14] M. PUTINAR,Hyponormal operators are subscalar, J. Operator Theory12(1984), 385-395. · Zbl 0573.47016
[15] H. RADJAVI ANDP. ROSENTHAL,Invariant subspaces, Springer-Verlag, 1973. · Zbl 0269.47003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.