A lower bound for the spectral radius

Abstract

We prove an inequality for a problem of Carathéodory type: given <italic>n</italic> inner functions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m 1 comma m 2 comma ellipsis comma m Subscript n Baseline"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>m</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>m</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mo>…<!-- … --></mml:mo> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>m</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{m_1},{m_2}, \ldots ,{m_n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, to find the smallest norm of an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript normal infinity"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>H</mml:mi> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{H^\infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> function such that the first <italic>n</italic> terms of its power series coincide with those of the product <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m 1 midline-horizontal-ellipsis m Subscript n"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>m</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mo>⋯<!-- ⋯ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>m</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{m_1} \cdots {m_n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. As an application, we obtain a lower bound for the spectral radius of an <italic>n</italic>-dimensional operator on Hilbert space in terms of its norm and the norm of its <italic>n</italic>th power.

Locations

• Proceedings of the American Mathematical Society - View - PDF