scholarly journals The minimal normal extension problem for subnormal operators

1986 ◽  
Vol 65 (3) ◽  
pp. 314-338 ◽  
Author(s):  
James Dudziak
Keyword(s):  
Extension Problem ◽  
Normal Extension ◽  
Subnormal Operators ◽  
Minimal Normal Extension

scholarly journals The spectrum of orthogonal sums of subnormal pairs

1988 ◽  
Vol 30 (1) ◽  
pp. 11-15 ◽  
Author(s):  
K. Rudol
Keyword(s):  
Invariant Subspace ◽  
Spectral Theory ◽  
Existence And Uniqueness ◽  
Normal Extension ◽  
Spectral Mapping Theorem ◽  
Spectral Mapping ◽  
Basic Properties ◽  
Subspace Theorem ◽  
Subnormal Operators ◽  
Minimal Normal Extension

This note provides yet another example of the difficulties that arise when one wants to extend the spectral theory of subnormal operators to subnormal tuples. Several basic properties of a subnormal operator Y remain true for tuples; e.g. the existence and uniqueness of its minimal normal extension N, the spectral inclusion σ(N)⊂ σ(Y)-proved for n-tuples in [4] and generalized to infinite tuples in [5]. However, neither the invariant subspace theorem nor the spectral mapping theorem in the “strong form” as in [3] is known so far for subnormal tuples.

Lifting the Commutant of a Subnormal Operator

1979 ◽  
Vol 31 (1) ◽  
pp. 148-156 ◽  
Author(s):  
Robert F. Olin ◽  
James E. Thomson
Keyword(s):  
Hilbert Space ◽  
Subnormal Operator ◽  
Basic Material ◽  
Normal Extension ◽  
Subnormal Operators ◽  
Double Commutant ◽  
Minimal Normal Extension

Let S be a subnormal operator on a Hilbert space ℋ and let N be its minimal normal extension on the Hilbert space ℋ. (We refer the reader to [5, 15] for the basic material on subnormal operators.) Denote the commutant and double commutant of an operator T by ﹛T﹜’ and ﹛T﹜”, respectively.

Spectral Inclusion and C.N.E.

1982 ◽  
Vol 34 (4) ◽  
pp. 883-887 ◽  
Author(s):  
A. R. Lubin
Keyword(s):  
Hilbert Space ◽  
Linear Operators ◽  
Normal Extension ◽  
Unitary Equivalence ◽  
Bounded Linear ◽  
Multi Index ◽  
Reducing Subspace ◽  
Normal Operators ◽  
Image Position ◽  
Minimal Normal Extension

1. An n-tuple S = (S1, …, Sn) of commuting bounded linear operators on a Hilbert space H is said to have commuting normal extension if and only if there exists an n-tuple N = (N1, …, Nn) of commuting normal operators on some larger Hilbert space K ⊃ H with the restrictions Ni|H = Si, i = 1, …, n. If we takethe minimal reducing subspace of N containing H, then N is unique up to unitary equivalence and is called the c.n.e. of S. (Here J denotes the multi-index (j1, …, jn) of nonnegative integers and N*J = N1*jl … Nn*jn and we emphasize that c.n.e. denotes minimal commuting normal extension.) If n = 1, then S1 = S is called subnormal and N1 = N its minimal normal extension (m.n.e.).

Weighted shifts and commuting normal extension

1979 ◽  
Vol 27 (1) ◽  
pp. 17-26 ◽  
Author(s):  
Arthur Lubin
Keyword(s):  
Normal Extension ◽  
Weighted Shifts ◽  
Subnormal Operators ◽  
Normal Operators

AbstractThe main result of this paper shows that the existence of commuting normal extension (c.n.e.) for an arbitrary family of commuting subnormal operators can be determined by considering appropriate families of multivariable weighted shifts. In proving this some known criteria for c.n.e. are generalized. It is also shown that a family of jointly quasi-normal operators has c.n.e.

scholarly journals Normal Extensions of Representations of Abelian Semigroups

2016 ◽  
Vol 59 (3) ◽  
pp. 564-574
Author(s):  
Boyu Li
Keyword(s):  
Normal Extension ◽  
Abelian Semigroup ◽  
Ordered Semigroups ◽  
Normal Representation ◽  
Subnormal Operators ◽  
Abelian Lattice

AbstractA commuting family of subnormal operators need not have a commuting normal extension. We study when a representation of an abelian semigroup can be extended to a normal representation, and show that it suffices to extend the set of generators to commuting normals. We also extend a result due to Athavale to representations on abelian lattice ordered semigroups.

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