A GEOMETRIC SPECTRAL THEOREM

1986 ◽  
Vol 37 (3) ◽  
pp. 263-277 ◽  
Author(s):  
YAAKOV FRIEDMAN ◽  
BERNARD RUSSO
Keyword(s):  
Spectral Theorem

A note on normal operators

1963 ◽  
Vol 59 (4) ◽  
pp. 727-729 ◽  
Author(s):  
S. J. Bernau ◽  
F. Smithies
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Adjoint Operator ◽  
Spectral Theorem ◽  
Unitary Operators ◽  
Unitary Space ◽  
Bounded Linear ◽  
Finite Dimensional ◽  
Normal Operators ◽  
Finite Dimensional Case

We recall that a bounded linear operator T in a Hilbert space or finite-dimensional unitary space is said to be normal if T commutes with its adjoint operator T*, i.e. TT* = T*T. Most of the proofs given in the literature for the spectral theorem for normal operators, even in the finite-dimensional case, appeal to the corresponding results for Hermitian or unitary operators.

scholarly journals The spectral theorem for well-bounded operators

1993 ◽  
Vol 54 (3) ◽  
pp. 334-351 ◽  
Author(s):  
Ian Doust ◽  
Qiu Bozhou
Keyword(s):  
Functional Calculus ◽  
Continuous Functions ◽  
Weak Compactness ◽  
Compact Interval ◽  
Spectral Theorem ◽  
Type B ◽  
Bounded Operators ◽  
Absolutely Continuous Functions ◽  
Absolutely Continuous

AbstractWell-bounded operators are those which possess a bounded functional calculus for the absolutely continuous functions on some compact interval. Depending on the weak compactness of this functional calculus, one obtains one of two types of spectral theorem for these operators. A method is given which enables one to obtain both spectral theorems by simply changing the topology used. Even for the case of well-bounded operators of type (B), the proof given is more elementary than that previously in the literature.

A noncommutative spectral theorem for operator entried companion matrices

1981 ◽  
Vol 10 (1) ◽  
pp. 45-51
Author(s):  
John de Pillis ◽  
Michael Neumann
Keyword(s):  
Spectral Theorem ◽  
Companion Matrices

scholarly journals From Freudenthal’s spectral theorem to projectable hulls of unital Archimedean lattice-groups, through compactifications of minimal spectra

2018 ◽  
Vol 30 (2) ◽  
pp. 513-526 ◽  
Author(s):  
Richard N. Ball ◽  
Vincenzo Marra ◽  
Daniel McNeill ◽  
Andrea Pedrini
Keyword(s):  
Explicit Construction ◽  
Continuous Functions ◽  
Spectral Theorem ◽  
Characteristic Functions ◽  
Prime Ideals ◽  
Minimal Extension ◽  
Strong Unit ◽  
Fundamental Relationship ◽  
Continuous Surjection ◽  
Archimedean Lattice

AbstractWe use a landmark result in the theory of Riesz spaces – Freudenthal’s 1936 spectral theorem – to canonically represent any Archimedean lattice-ordered groupGwith a strong unit as a (non-separating) lattice-group of real-valued continuous functions on an appropriateG-indexed zero-dimensional compactification{w_{G}Z_{G}}of its space{Z_{G}}ofminimalprime ideals. The two further ingredients needed to establish this representation are the Yosida representation ofGon its space{X_{G}}ofmaximalideals, and the well-known continuous surjection of{Z_{G}}onto{X_{G}}. We then establish our main result by showing that the inclusion-minimal extension of this representation ofGthat separates the points of{Z_{G}}– namely, the sublattice subgroup of{\operatorname{C}(Z_{G})}generated by the image ofGalong with all characteristic functions of clopen (closed and open) subsets of{Z_{G}}which are determined by elements ofG– is precisely the classical projectable hull ofG. Our main result thus reveals a fundamental relationship between projectable hulls and minimal spectra, and provides the most direct and explicit construction of projectable hulls to date. Our techniques do require the presence of a strong unit.

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