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Dilations, Functoinal Model And A Complete Unitary Invariant Of A r-contraction.

Electronic Theses of Indian Institute of Science

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Title Dilations, Functoinal Model And A Complete Unitary Invariant Of A r-contraction.
 
Creator Pal, Sourav
 
Subject Set Contraction
Hilbert Space
Contraction Operators
Functional Analysis
Operator Theory
Dilation Theory
r-unitaries
r-isometries
r-contraction
Mathematics
 
Description A pair of commuting bounded operators (S, P) for which the set
r = {(z 1 +z 2,z 1z 2) : |z 1| ≤1, |z 2| ≤1} C2 is a spectral set, is called a r-contraction in the literature. For a contraction P and a bounded commutant S of P, we seek a solution of the operator equation
S –S*P = (I –P*P)½ X(I –P*P)½
where X is a bounded operator on Ran(I – P*P)½ with numerical radius of X being not
greater than 1. We show the existence and uniqueness of solution to the operator equation above when (S,P) is a r-contraction. We call the unique solution, the fundamental operator of the r-contraction (S,P). As the title indicates, there are three parts of this thesis and the main role in all three parts is played by the fundamental operator. The existence of the fundamental operator allows us to explicitly construct a r-isometric dilation of a r-contraction (S,P), whereas its uniqueness guarantees the uniqueness of the minimal r-isometric dilation. The fundamental operator helps us to produce a genuine functional model for pure r-contractions. Also it leads us to a complete unitary invariant for pure r-contractions. We decipher the structures of r-isometries and r-unitaries by characterizing them in several different ways. We establish the fact that for every pure r-contraction (S,P), there is a bounded operator C with numerical radius being not greater than 1 such that S = C + C* P. When (S,P) is a r-isometry, S has the same form where P is an isometry commuting with C and C*. Also when (S,P) is a r-unitary, S has the same form too with P and C being commuting unitaries. Examples of r-contractions on reproducing kernel Hilbert spaces and their dilations are discussed.
 
Contributor Bhattacharyya, Tirthankar
 
Date 2013-08-02T09:56:45Z
2013-08-02T09:56:45Z
2013-08-02
2011-11
 
Type Thesis
 
Identifier http://etd.iisc.ernet.in/handle/2005/2182
http://etd.ncsi.iisc.ernet.in/abstracts/2790/G25106-Abs.pdf
 
Language en_US
 
Relation G25106