特別なシンボルに対する有限階中間ハンケル作用素 4

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This paper extends some results of the previous papers [1, 3, 4, 5]. Let dμ=dσ(r)dθ be a Borel measure on the open unit disc D where dσ (r) is a positive measure on [0, 1] with dσ([0, 1])=1/ 2π and dθ is the Lebesgue measure on 〓D. We assume that L^2_a is a closed subspace of L^2 whose element is analytic on D as in [3]. We suppose M is a closed subspace in L^2 which is invariant under the multiplication by the coordinate function z. Let P^M denotes the orthogonal projection from L^2 onto M. For φ in L^∞ (D, dμ) , the intermediate Hankel operator H^M_φ is defined by H^M_φ f= (I-P^M) (φf ) for f in L^2_a. In the paper [1], we give three necessary and sufficient conditions that H^M_φ is of finite rank in case M⊇zL^2_a, when φ is in L^∞. In this paper, we give the same conditions in that H^M_φ is of finite rank in case M⊆L^2_a, when φ is in L^∞_a (D, dμ). As a result, we show that if M is of finite codimension in L^2_a then there does not exist a finite rank H^M_φ except H^M_φ=0.
旭川工業高等専門学校の論文
2005-03-05