【摘要】：在目前的數(shù)學(xué)領(lǐng)域中,對(duì)偶Toeplitz算子理論方面的內(nèi)容多是圍繞在Hardy空間、Bergman空間,甚至是調(diào)和Bergman空間上,而調(diào)和Hardy空間上的理論則相對(duì)少很多。本篇論文就是著眼于調(diào)和Hardy空間上對(duì)偶Toeplitz算子性質(zhì)的研究。論文一共分為四部分。第一部分的緒論部分首先介紹了一下目前的研究情況,重點(diǎn)介紹了多圓環(huán)T~n上的調(diào)和Hardy空間h~2(T~n)=H~2(T~n)+H~2(T~n)的定義。并且給出了作用在空間h~2(T~n)的補(bǔ)空間上的對(duì)偶Toeplitz算子的定義：其中Q=I-P,而P表示從空間L2(T~n)到空間h~2(T~n)上的正交投影。還通過(guò)空間L2(T~n)上乘法算子Mφ,得出了等式這個(gè)等式在后面文章的證明中經(jīng)常用到。第二部分的核心內(nèi)容是空間h~2(T~n)上的譜包含定理：如果φ∈L∞(T~n),則R(φ)(?)σ(Sφ)。完成了定理的證明之后又介紹了幾條由此得出的空間h~2(T~n)上的常用推論。例如若算子Sφ是自伴的,當(dāng)且僅當(dāng)φ是實(shí)值函數(shù)。有了譜包含定理之后,第三部分開(kāi)始研究h~2(T~n)上的對(duì)偶Toeplitz算子Sφ的交換性。通過(guò)簡(jiǎn)單的實(shí)例,就可以清晰地知道空間h~2(T~n)的調(diào)和性在交換性等性質(zhì)方面有著非常重要的作用,因此并不能得出適合所有算子Sφ的結(jié)論。本文只研究了n=2,并且對(duì)偶Toeplitz算子Sφ的符號(hào)函數(shù)具有如下特殊形式的情況下的交換性質(zhì)：其中f,g∈H∞(D2),z,w∈T,mi,ni∈N,i=1,2.而最后一部分的算子Sφ的半交換性的研究也是在與第三部分相同的前提下進(jìn)行的,通過(guò)對(duì)參數(shù)m_i,n_i的分情況討論得出結(jié)論。
[Abstract]:In the field of mathematics at present, the theory of dual Toeplitz operators is mostly centered on Hardy space, Bergman space, and even harmonic Bergman space, while the theory of harmonic Hardy space is relatively few. This paper focuses on the study of the properties of dual Toeplitz operators on harmonic Hardy spaces. The paper is divided into four parts. The introduction of the first part first introduces the present research situation, and focuses on the definition of harmonic Hardy space Hn2 (Tnn) = Hn2 (Tnn) on the polycircular ring Tnn. We also give the definition of dual Toeplitz operator acting on the complementary space of space HG 2 (Tn), where QG I-P, and P denotes the orthogonal projection from space L2 (Tn) to space HG 2 (Tn). By means of the multiplication operator M 蠁 on the space L2 (Tn), it is obtained that the equation is often used in the proof of the following papers. The core of the second part is the spectral inclusion theorem on the space H ~ (2) (T _ (n): if 蠁 鈭，

本文編號(hào)：2384606