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&#8722;1} = q_{ij} + ba_{ij}
a_{2i&#8722;1,2j} = v_{ij} + ca_{ij}
a_{2i&#8722;1,2j&#8722;1} = w_{ij} + da_{ij}
所有 i, j &#8712; 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 &#8712;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&#8722;1} = q_{ij} + ba_{ij}
a_{2i&#8722;1,2j} = v_{ij} + ca_{ij}
a_{2i&#8722;1,2j&#8722;1} = w_{ij} + da_{ij}
for all i, j &#8712; Z, where (p_{ij}), (q_{ij}), (v_{ij}), (w_{ij}) are bounded matrices on l^2(Z) and a, b, c, d &#8712;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

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