By Ronald G. Douglas, Steven G. Krantz, Eric T. Sawyer, Sergei Treil, Brett D. Wick

The function of the corona workshop used to be to contemplate the corona challenge in either one and a number of other complicated variables, either within the context of functionality idea and harmonic research in addition to the context of operator idea and sensible research. It used to be held in June 2012 on the Fields Institute in Toronto, and attended via approximately fifty mathematicians. This quantity validates and commemorates the workshop, and files many of the principles that have been constructed within.

The corona challenge dates again to 1941. It has exerted a strong impact over mathematical research for almost seventy five years. there's fabric to assist deliver humans up to the mark within the newest rules of the topic, in addition to historic fabric to supply history. relatively noteworthy is a heritage of the corona challenge, authored by way of the 5 organizers, that gives a special glimpse at how the matter and its many alternative options have developed.

There hasn't ever been a gathering of this type, and there hasn't ever been a quantity of this sort. Mathematicians—both veterans and newcomers—will make the most of examining this booklet. This quantity makes a distinct contribution to the research literature and may be a worthy a part of the canon for a few years to come.

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**Extra resources for The Corona Problem: Connections Between Operator Theory, Function Theory, and Geometry**

**Sample text**

M is called projective if there exists an R-module N and an integer d that M ˚ N Š Rd . 0 such In terms of matrices, the ring R is projective free iff every square idempotent matrix F is conjugate (by an invertible matrix) to a matrix of the form Ä Ik 0 ; 0 0 see [Co1, Prop. 6]. From the matricial definition, it follows that any field k is projective free, since matrices F satisfying F 2 D F are diagonalizable over k. Quillen and Suslin (see [La]) proved, independently, that the polynomial ring over a projective free ring is again projective free.

LOMI) 113 (1981), 178–198, 267 (Russian, with English summary). Investigations on linear operators and the theory of functions, XI. 66. S. Treil, An operator Corona theorem, Indiana Univ. Math. J. 53 (2004), no. 2640. MR2106344 (2005j:30067) 67. ——, The problem of ideals of H 1 : beyond the exponent 3/2, J. Funct. Anal. 253 (2007), no. 1, 220–240. 68. ——, Lower bounds in the matrix Corona theorem and the codimension one conjecture, Geom. Funct. Anal. 14 (2004), no. 5, 1118–1133. A History of the Corona Problem 29 69.

Z/ for all z 2 D. Seemingly much stronger, this problem is equivalent to the operator corona problem for X1 ; X2 being separable Hilbert cases. This result, known as the Tolokonnikov lemma, is proved in full generality by S. Treil in [Tr7]. ) In general, the operator completion problem is much more involved. Some 1 sufficient conditions for its solvability for Hcomp spaces were given by the author in [Br8, Th. 3]. X2 ; X1 ˚ Y //, respectively. For instance, this is valid in one of the following cases (see [Br10, Mi]): (a) dimC Y < 1; (b) X2 is isomorphic to a Hilbert space or c0 or one of the spaces `p , 1 Ä p Ä 1; (c) X2 is isomorphic to one of the spaces Lp Œ0; 1, 1 < p < 1, or C Œ0; 1 and X1 is not isomorphic to X2 .