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論文名稱 Title |
λ-托普立茲算子之 Fredholm 譜 Fredholm spectra of λ-Toeplitz operators |
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系所名稱 Department |
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畢業學年期 Year, semester |
語文別 Language |
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學位類別 Degree |
頁數 Number of pages |
14 |
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研究生 Author |
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指導教授 Advisor |
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召集委員 Convenor |
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口試委員 Advisory Committee |
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口試日期 Date of Exam |
2011-01-14 |
繳交日期 Date of Submission |
2011-07-25 |
關鍵字 Keywords |
合成算子、λ-托普立茲算子、托普立茲算子 Toeplitz operator, composition operator, λ-Toeplitz operator |
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統計 Statistics |
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中文摘要 |
中文摘要 令λ是在一個封閉圓盤上的一個複數,而H是一個以正交基底所構成的可分希爾伯特空間,也就是,基底為 ε={ en:n= 0,1,2,...}的形式。一個作用在H的有界算子若滿足以下的定義運算<Tem+1 ,en+1 >= λ<Tem ,Ten >則稱之為λ-托普立茲算子(此處<•, •>表示作用在H上的內積)。若函數φ可表示為上述正交基底之線性組合, 且其係數為 an=<Te0 ,en >, n≥ 0,和 an=<Telnl ,e0 >, n<0,這就稱為T的symbol。這個主題很自然地來自一個特殊的算子方程式 S*AS=λA+B﹐ S是作用在H空間上的一個shift算子, 其中A矩陣為一組特殊聯立方成組的解。顯然的著名的托普立茲算子即為S*AS=λA的解,當S為一個單方面的shift算子時。在此篇文章中,首先我們會回顧λ-托普立茲算子和托普立茲算子之間的相似性和差異性。最後我們會將著名的Coburn’s characterization其在托普立茲算子的essential spectrum的應用上(或者在spectrum的應用上)推廣至λ-托普立茲算子。 |
Abstract |
Abstract Let λ be a complex number in the closed unit disc , and H be a separable Hilbert space with the orthonormal basis, say,ε= {en : n =0 , 1 , 2…}. A bounded operator T on H is called a λ-Toeplitz operator if <Tem+1 , en+1> =λ <Tem , en> (where <•,•> is the inner product on H).If the function φ can be represented as a linear combination of the above orthonormal basis with the coefficients an=<Te0 ,en >, n≥ 0,and an=<Telnl ,e0 >, n<0, then we call this the symbol of T . The subject arises naturally from a special case of the operator equation S*AS =λA + B; where S is a shift on H , and in this operator equation the matrix A can solve a special set of simultaneous equations. It is also clear that the well-known Toeplitz operators are precisely the solutions of S*AS = A, when S is the unilateral shift.In this paper,we will review the similarities and differences between λ-Toeplitz operators and Toeplitz operators. The main purpose is to generalize the well-known Coburn's characterization for the essential spectrum(or,the same in this case,spectrum)for Toeplitz operators to λ-Toeplitz operators. |
目次 Table of Contents |
Contents 中文摘要.......... i Abstract........... ii 1. Introduction ..........1 2. Some elementary properties of the λ-Toeplitz operators ..................2 3. The essential spectrum of λ-Toeplitz operators.................. 4 References.............. 6 |
參考文獻 References |
References [1] R. G. Douglas, Banach Algebra Techniques in Operator Theory, 2nd ed.,Springer-Verlag, New York, 1998. [2] M. C. Ho, Adjoints of slant Toeplitz operators II, Integral Equations and Operator Theory, 41, 2001, pp.179-188. [3] M. C. Ho and M. M. Wong, Operators that commute with slant Toeplitz operators, Applied Math. Research eXpress, 2008, Article ID abn003, 20pages, doi:10.1093/amrx/abn003. [4] M. C. Ho, Solutions to a dyadic recurrent system and certain action on B(H) induced by shifts, submitted. [5] M. C. Ho, A simple comparison of the Toeplitz and the λ-Toeplitz operators, submitted. [6] M. T. Jury, The Fredholm index for elements of the Toeplitz-Composition C*-algebra, Integral Equations and Operator Theory, 58, 2007, pp.341-362. [7] P. Walters, An Introduction to Ergodic Theory, Graduate Text in Mathematics 79, Springer-Verlag, New York, 1982. [8] A. Wintner, Zur theorie der beschrankten bilinear formen, Math Z., 30,1929, pp.228-282. |
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