行政院國家科學委員會專題研究計畫 成果報告
函數代數上的位移算子理論(2/2)
計畫類別： 個別型計畫 計畫編號： NSC92-2115-M-110-003- 執行期間： 92 年 08 月 01 日至 93 年 07 月 31 日 執行單位： 國立中山大學應用數學系(所) 計畫主持人： 黃毅青 報告類型： 完整報告 報告附件： 出席國際會議研究心得報告及發表論文 處理方式： 本計畫可公開查詢中 華 民 國 93 年 12 月 10 日
行政院國家科學委員會專題研究-完整報告
函數代數上的位移算子理論(2/2)
計畫編號：
92-2115-M-110-003執行期限：92 年 08 月 01 日至 93 年 07 月 31 日
主持人：黃毅青 國立中山大學應用數學系
一、中文摘要 本計畫為一兩年期計畫，主題是函數代數上位移算子之研究。 我們將建立一個巴拿赫空間上的位移算子理論，而且這並不依賴於基底的存在與 否。在計畫的第一年期，定義在連續函數空間 C0(X)上保距和保斥性位移算子是主要的 研究對像。在計畫的第二年期，一個完整的位移算子理論將被建立。我們相信這將對碎 形、小波和其他具有泛函分析背景的理論或應用問題的研究有所幫助。 關鍵詞：C*-代數，Feldholm 算子，Gleason-Kahane-Zelazko 定理，有限餘維子空間。 AbstractIn this two year project, we shall study shift operators on function algebras.
We are interested in generalizing the notion of shifts and quasi-shifts to Banach spaces in a basis free setting. In the first year, we shall study isometric and disjointness preserving shift operators on the continuous function spaces C0(X). In the second year, we expect to obtain a comprehensive theory of shift operators. This should be helpful in the study in fractals, wavelet and other pure and applied problems where a functional analsysis setting is allowed.
Keywords: Shift operators, function algebras, isometry, disjointness preserving.
二、緣由與目的
Let T be a linear isometry from an infinite dimensional separable Hilbert space H into H of finite corank n. The von Neumann--Wold Decomposition Theorem (see, e.g., [6, p.112]) states that T can be written as a direct sum of a unitary and a product of n copies of the unilateral shift. More precisely, Hu =∩m TmH is a reducing subspace of T. Its orthogonal
complement Hs=HӨ Hu is the infinite orthogonal sum of TmN, where N=H TH is of
dimension n. Now, T|Hu is a unitary and T| Hs shifts each n-dimensional subspace TmN onto Tm+1N for m = 0, 1, 2 ,... In this sense, we may call T an isometric quasi-n-shift on H.
Generalizing a notion of Crownover, we call a (necessarily bounded) linear operator S from a Banach space E into E an n-shift if S is injective, has closed range of codimension n,
if S satisfies all conditions but not necessarily the last.
Let X and Y be locally compact Hausdorff spaces. For a linear isometry T : C0(X) →
C0(Y) of finite corank, there is a cofinite subset Y1 of Y such that Tf|Y1=h· f ◦φ is a weighted composition operator and X is homeomorphic to a quotient space of Y1 modulo a finite subset. When X=Y, such a T is an isometric quasi-n-shift on C0(X). In this case, the action of T can be implemented as a shift on a tree-like structure, called a T-tree, in M(X) with exactly n joints. Here, M(X) is the dual space of C0(X) consisting of bounded regular Borel measures on X. The
T-tree is total in M(X) when T is a shift. With these tools, we can analyze the structure of T.
We show that every disjointness preserving (quasi-)n-shift on c0 can always be written as a product of n (quasi-)shifts. Although it is not the case for general C0(X) as shown by our counter examples, we can do so after dilation.
Some useful results in this direction can be found in the papers mentioned in the reference section. We expect we can make some progress in this topic, too.
三、結果與討論
經過了系統的研究，在多名研究生的參與下，基本完成了準備工作，為以後的工作 作好準備。並且完成了以下四篇論文[1, 6, 16, 17] ，其主要結果如下：
For a linear isometry T: C0(X) Æ C0(Y) with finite corank, we show that there exists a cofinite subset Y1 of Y on which T can be written as a weighted composition map. They also prove that X must be homeomorphic to a quotient space of Y1 modulo a finite subset and provide a useful description of the range of T.
Assume X=Y, which is to say that T is an isometric quasi-n-shift. They discuss the structure of the range spaces of the powers Tk of T and show that T can be implemented by a shift on a countable set with a tree-like structure, which they call T-tree, with exactly n joints in the dual space M(X) of C0(X). If T is an isometric n-shift, then the T-tree is total in M(X). With these tools, they obtain, among other results, the following: X must be separable if C0(X) admits a disjointness-preserving isometric n-shift.
In other coming papers, we will discuss the disjointness preserving quasi-n-shifts. A necessary conditions that a locally compact Hausdorff space X admits a disjointness preserving shift is that the first homology group H(X) of X satisfies the condition H(X)/Z ≈
H(X). This rules out the possibility that a compact manifold admitting such a shift.
To extend the theory further, we consider to remove the continuity of the disjointness preserving shifts. In [1], we work on the structure of unbounded disjointness preserving linear functionals on C0(X). We find that these functionals arise from prime ideals of C0(X). More precisely, if I is an prime ideal of C0(X), then any linear lifting to C0(X) of a non-zero linear functional of C0(X)/I is disjointness preserving. It is shown that I is contained and dense in a unique maximal ideal M of C0(X). So the lifting is bounded if and only if it is zero on M. This gives us a lot of freedom to construct unbounded disjointness preserving linear
functionals. We hope this machinery can help on studying the unbounded shift on C0(X). 四、計畫成果自評 本年度為此二年計劃的第二年。我們已經獲得了預期的成果和完成了以上四篇論 文，其中第四篇論文是在本年度完成。預計通過日後的繼續研究，將有更多更好的結果。 由此推動國內外本門學科之發展。本人、研究助理及博碩士生得到充份的研究訓練，將 發掘新的研究題材。本研究可視為國人探索位移算子一般理論的準備。 五、參考文獻
1. Lawrence G. Brown and Ngai-Ching Wong, “Unbounded disjointness preserving linear functionals”, Monatshefte für Mathematik, 141 (2004), no. 1, 21 - 32.
2. W. Holsztynski, Continuous mappings induced by isometries of spaces of continuous functions, Studia Math., 26 (1966), 133-136.
3. J-S. Jeang and N-C. Wong, Weighted composition operators of C0(X)'s, J. Math. Anal.
Appl., 201 (1996), 981--993.
4. J. Araujo and J. J. Font, Codimension 1 linear isometries on function algebras, Proc. Amer. Math. Soc., 127 (1999), 2273--2281.
5. J. Araujo and J. J. Font, Isometric shifts and metric spaces, to appear in Monatsh. Math. 6. Li-Shu Chen, Jyh-Shyang Jeang and Ngai-Ching Wong, “Disjointness preserving shifts
on C0(X)'s”, submitted to Studia Math.
7. J. B. Conway, A course in operator theory, American Mathematical Society, Providence, Rhode Island 2000.
8. R. M. Crownover, Commutants of shifts on Banach spaces, Michigan Math. J., 19 (1972), 233--247.
9. F. O. Farid and K. Varadarajan, Isometric shift operators on C(X), Can. J. Math., 46 (1994), 532--542.
10. J. J. Font, Isometries between function algebras with finite codimensional range, Manuscripta Math., 100 (1999), 13--21.
11. A. Gutek, D. Hart, J. Jamison and M. Rajagopalan, Shift operators on Banach spaces, J. Funct. Anal., 101 (1991), 97--119.
12. R. Haydon, Isometric shifts on C(K), J. Funct. Anal., 135 (1996), 157--162. 13. J. R. Holub, On shift operators, Canad. Math. Bull., 31 (1988), 85--94.
14. W. Holsztynski, Continuous mappings induced by isometries of spaces of continuous functions, Studia Math., 26 (1966), 133-136.
Appl., 201 (1996), 981--993.
16. J.-S. Jeang and N.-C. Wong, ``Isometric shifts on C0(X)'s'', J. Math. Anal. Appl., vol. 274, no. 2 (2002), 772-787.
17. J.-S. Jeang and N.-C. Wong, Disjointness preserving Fredholm linear operators of C0(X), J. Operator Theory, vol. 49, no. 1 (2003), 61-75.
18. M. Rajagopalan and K. Sundaresan, Backward shifts on Banach spaces C(X), J. Math. Anal. Appl., 202 (1996), 485--491.
19. M. Rajagopalan and K. Sundaresan, Backward shifts on Banach spaces C(X) II, in ``Proceedings of the Tennessee Topology Conference,'' World Scientific, 1996, 199--205. 20. M. Rajagopalan, T. M. Rassias and K. Sundaresan, Generalized backward shifts on
