The essential norms of composition operators between generalized Bloch spaces in the polydisc and their applications

• Main Authors: Zhou Zehua, Liu Yan
• Format: Article
• Language: English
• Published: SpringerOpen 2006-01-01
• Series: Journal of Inequalities and Applications
• Online Access: http://www.journalofinequalitiesandapplications.com/content/2006/90742
• ISSN: 1025-5834; 1029-242X

<p/> <p>Let <inline-formula><graphic file="1029-242X-2006-90742-i1.gif"/></inline-formula> be the unit polydisc of <inline-formula><graphic file="1029-242X-2006-90742-i2.gif"/></inline-formula> and <inline-formula><graphic file="1029-242X-2006-90742-i3.gif"/></inline-formula> a holomorphic self-map of <inline-formula><graphic file="1029-242X-2006-90742-i4.gif"/></inline-formula>. <inline-formula><graphic file="1029-242X-2006-90742-i5.gif"/></inline-formula>, <inline-formula><graphic file="1029-242X-2006-90742-i6.gif"/></inline-formula> and <inline-formula><graphic file="1029-242X-2006-90742-i7.gif"/></inline-formula> denote the <inline-formula><graphic file="1029-242X-2006-90742-i8.gif"/></inline-formula>-Bloch space, little <inline-formula><graphic file="1029-242X-2006-90742-i9.gif"/></inline-formula>-Bloch space, and little star <inline-formula><graphic file="1029-242X-2006-90742-i10.gif"/></inline-formula>-Bloch space in the unit polydisc <inline-formula><graphic file="1029-242X-2006-90742-i11.gif"/></inline-formula>, respectively, where <inline-formula><graphic file="1029-242X-2006-90742-i12.gif"/></inline-formula>. This paper gives the estimates of the essential norms of bounded composition operators <inline-formula><graphic file="1029-242X-2006-90742-i13.gif"/></inline-formula> induced by <inline-formula><graphic file="1029-242X-2006-90742-i14.gif"/></inline-formula> between <inline-formula><graphic file="1029-242X-2006-90742-i15.gif"/></inline-formula> ( <inline-formula><graphic file="1029-242X-2006-90742-i16.gif"/></inline-formula> or <inline-formula><graphic file="1029-242X-2006-90742-i17.gif"/></inline-formula>) and <inline-formula><graphic file="1029-242X-2006-90742-i18.gif"/></inline-formula> ( <inline-formula><graphic file="1029-242X-2006-90742-i19.gif"/></inline-formula> or <inline-formula><graphic file="1029-242X-2006-90742-i20.gif"/></inline-formula>). As their applications, some necessary and sufficient conditions for the (bounded) composition operators <inline-formula><graphic file="1029-242X-2006-90742-i21.gif"/></inline-formula> to be compact from <inline-formula><graphic file="1029-242X-2006-90742-i22.gif"/></inline-formula> ( <inline-formula><graphic file="1029-242X-2006-90742-i23.gif"/></inline-formula> or <inline-formula><graphic file="1029-242X-2006-90742-i24.gif"/></inline-formula>) into <inline-formula><graphic file="1029-242X-2006-90742-i25.gif"/></inline-formula> ( <inline-formula><graphic file="1029-242X-2006-90742-i26.gif"/></inline-formula> or <inline-formula><graphic file="1029-242X-2006-90742-i27.gif"/></inline-formula>) are obtained.</p>