Spectrum of a composition operator

Abstract

<italic>A composition operator</italic> is a linear operator induced on a subspace of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K Superscript upper X"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>K</mml:mi> <mml:mi>X</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{K^X}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by a point transformation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi"> <mml:semantics> <mml:mi> ϕ </mml:mi> <mml:annotation encoding="application/x-tex">\phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on a set <italic>X</italic> (where <italic>K</italic> denotes the scalar field) by the formula <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T f left-parenthesis x right-parenthesis equals f ring phi left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> <mml:mo> ∘ </mml:mo> <mml:mi> ϕ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">Tf(x) = f \circ \phi (x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Familiar examples include translation operators on the real line and on topological groups, analytic functions which preserve the class of harmonic functions (and Green’s functions), ergodic transformations which induce unitary operators on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L squared"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{L^2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , shift and weighted shift operators. The spectrum, approximate point spectrum, and point spectrum of an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript p"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{L^p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -composition operator have circular symmetry about 0, except on the unit circle, where they form unions of subgroups; certain consequences are derived from this.