給定 λ 為單位閉圓裡的複數，H是可分離的希爾伯特空間，且 ε={e_n:n=0,1,2,…} 為H一單範正交基底。在H裡，一有界算子T若滿足< Te_{m+1},Te_{n+1} >=λ< Te_m,Te_n > ( <•,•> 為H裡的內積 )，則稱T為 λ-托普立茲算子。這主題源自於最近研究一個特例算子方程式S*AS = λA + B, 這裡的S是在H裡的shift, 對於在l^2(Z)裡找一有界矩陣 (a_{ij})滿足以下方程式扮演著不可或缺的角色
a_{2i,2j} = p_{ij} + aa_{ij}
a_{2i,2j−1} = q_{ij} + ba_{ij}
a_{2i−1,2j} = v_{ij} + ca_{ij}
a_{2i−1,2j−1} = w_{ij} + da_{ij}
所有 i, j ∈ Z, (p_{ij}), (q_{ij}), (v_{ij}), (w_{ij}) 皆為在 l^2(Z) 裡的有界矩陣且a, b, c, d ∈C。
當S是一unilateral shift，我們可以精確的得到所熟知的托普立茲算子為S*AS = A的解。這篇文章主要是討論 λ-托普立茲算子的一些基本性質，像是有界性和緊致性，並且比較相關結果與托普立茲算子差異。

Let λ be a complex number in the closed unit disc, and H be a separable Hilbert space with the orthonormal basis, say, ε={e_n:n=0,1,2,…}. A bounded operator T on H is called a λ-Toeplitz operator if < Te_{m+1},Te_{n+1} >=λ< Te_m,Te_n > (where <•,•> is the inner product on H).
The subject arises just recently from a special case of the operator equation S*AS = λA + B, where S is a shift on H, which plays an essential role in finding bounded matrix (a_{ij}) on l^2(Z) that solves the system of equations
a_{2i,2j} = p_{ij} + aa_{ij}
a_{2i,2j−1} = q_{ij} + ba_{ij}
a_{2i−1,2j} = v_{ij} + ca_{ij}
a_{2i−1,2j−1} = w_{ij} + da_{ij}
for all i, j ∈ Z, where (p_{ij}), (q_{ij}), (v_{ij}), (w_{ij}) are bounded matrices on l^2(Z) and a, b, c, d ∈C.
It is also clear that the well-known Toeplitz operators are precisely the solutions of S*AS = A, when S is the unilateral shift. The purpose of this paper is to discuss some basic topics, such as boundedness and compactness, of the λ-Toeplitz operators, and study the similarities and the differences with the corresponding results for the Toeplitz operators.

1. Introduction 3
2. Boundedness of T_{λ,φ} 6
3. Compactness of T_{λ,φ} 10

[1] A. Brown and P. R. Halmos, Algebraic properties of Toeplitz operators, J. reine angew. Math., 231, 1963, pp.89-102.
[2] M. D. Contreras and A. G. Hern′andaz-D′ıaz, Weighted composition operators between different Hardy spaces, Integral Equations and Operator Theory, 46, 2003, pp.165-188.
[3] M. D. Contreras and A. G. Hern′andaz-D′ıaz, Weighted composition operators on space of functions with derivatives in a Hardy space, J. Operator Theory, 52, 2004, pp.173-184.
[4] C. C. Cowen and E. Ko, Weighted composition operators on H^2, Preprint.
[5] G. Gunatillake, Spectrum of a compact weighted composition operator, Proc. AMS, 135, no. 2, 2007, pp.461-467.
[6] G. Gunatillake, Compact weighted composition operators on the Hardy space, Proc. AMS, 136, no. 8, 2008, pp.2895-2899.
[7] M. C. Ho, Adjoints of slant Toeplitz operators II, Integral Equations and Operator Theory, 41, 2001, pp.179-188.
[8] M. C. Ho and M. M. Wong, Operators that commute with slant Toeplitz operators, Applied Math. Research eXpress, 2008, Article ID abn003, 20 pages, doi:10.1093/amrx/abn003.
[9] M. C. Ho and C. M. Hsu, On infinite matrices on l^2(Z) generated dyadically by a 2 × 2 block, submitted.
[10] C. M. Hsu, On infinite matrices whose entries satisfying certain dyadic recurrent formula, master thesis, NSYSU, 2007.
[11] B. D. MacCluer and R. Zhao, Essential norms of weighted composition operators between Bloch-type spaces, Rocky Mountain J. Math., 33, no. 4, 2003.
[12] S. Ohno, Weighted composition operators between H^1 and the Bloch space, Taiwanese J.
Math., 5, no. 3, 2001, pp.555-563.