scholarly journals Connections of the Corona Problem with Operator Theory and Complex Geometry

2014 ◽  
pp. 47-68
Author(s):  
Ronald G. Douglas
Keyword(s):  
Operator Theory ◽  
Complex Geometry ◽  
Corona Problem

Von Neumann Operators in

1984 ◽  
Vol 27 (2) ◽  
pp. 146-156
Author(s):  
Karim Seddighi
Keyword(s):  
Operator Theory ◽  
Functional Calculus ◽  
Complex Geometry ◽  
Main Concern ◽  
Spectral Mapping ◽  
Special Situation ◽  
Von Neumann ◽  
Connected Open Subset ◽  
Spectral Set ◽  
Simply Connected

AbstractFor a connected open subset Ω of the plane and n a positive integer, let be the space introduced by Cowen and Douglas in their paper, “Complex geometry and operator theory”. Our main concern is the case n = 1, in which case we show the existence of a functional calculus for von Neumann operators in for which a spectral mapping theorem holds. In particular we prove that if the spectrum of , is a spectral set for T, and if , then σ(f(T)) = f(Ω)- for every bounded analytic function f on the interior of L, where L is compact, σ(T) ⊂ L, the interior of L is simply connected and L is minimal with respect to these properties. This functional calculus turns out to be nice in the sense that the general study of von Neumann operators in is reduced to the special situation where Ω is an open connected subset of the unit disc with .

scholarly journals Complex geometry and operator theory

1978 ◽  
Vol 141 (0) ◽  
pp. 187-261 ◽  
Author(s):  
M. J. Cowen ◽  
R. G. Douglas
Keyword(s):  
Operator Theory ◽  
Complex Geometry

ON AN OPERATOR THEORY APPROACH TO THE CORONA PROBLEM

2002 ◽  
Vol 34 (3) ◽  
pp. 369-373
Author(s):  
E. AMAR ◽  
C. MENINI
Keyword(s):  
Operator Theory ◽  
Negative Answer ◽  
The Other ◽  
Theory Approach ◽  
Corona Problem ◽  
Natural Approach ◽  
Counter Example ◽  
Other Hand ◽  
Commutant Lifting

This paper deals with an operator theory approach to the corona conjecture for H∞([ ]n). Treil gave a counter-example to this conjecture in the case where n = 1 for operator-valued functions; thus one might hope to use this to disprove the corona conjecture for H∞([ ]n) (for n [ges ] 2). This paper shows that this natural approach towards a negative answer fails. On the other hand, the second result here shows that ‘commutant lifting’ cannot be true for more than two contractions for any constant. This obstructs a natural attempted proof of the corona conjecture for H∞([ ]n) (for n [ges ] 2) by our previous result.

scholarly journals Complex geometry and operator theory

1977 ◽  
Vol 83 (1) ◽  
pp. 131-134 ◽  
Author(s):  
M. J. Cowen ◽  
R. G. Douglas
Keyword(s):  
Operator Theory ◽  
Complex Geometry
Sign in / Sign up

Export Citation Format

Share Document