# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
The essential norm of a composition operator. (English) Zbl 0642.47027

Let ${\Omega }\subset {ℂ}^{n}$ be a domain and ${\Phi }:{\Omega }\to {\Omega }$ a mapping. The operator $T:f\to f\circ {\Phi }$ is called a composition operator.

The subject of composition operators represents a fertile arena for the interaction of operator theory, hard analysis, and geometry. Only a few dozen papers have been written in the field so far, and these have been primarily concerned with function spaces on homogeneous domains - mainly balls and polydiscs. I would like to see the theory of composition operators developed on, say, strongly pseudoconvex domains in ${ℂ}^{n}·$ The opportunities to relate deep properties of Kähler geometry to deep properties of canonical operators seem manifest.

J. Shapiro is one of the foremost workers in the field of composition operators, and this paper represents a high point in the subject. He obtains a complete characterization of compact composition operators on ${H}^{2}\left(D\right)$, $D=\left\{z\in ℂ:|z|<1\right\}$, together with a number of interesting consequences for peak sets, essential norm of composition operators, etc.

I recommend this paper as a delightful introduction to an important topic that has not been sufficiently explored.

Reviewer: St.G.Krantz

##### MSC:
 47B38 Operators on function spaces (general) 46E25 Rings and algebras of continuous, differentiable or analytic functions 46E20 Hilbert spaces of continuous, differentiable or analytic functions 30D55 H (sup p)-classes (MSC2000)