Toeplitz Operators And Its Algebra On Two Kinds Of Function Spaces

In this paper, we mainly have discussed the compactness of Toeplitz operator with essential bound symbol on Bergman space and Dirichlet space respectively. Lastly we also have given the canonical decomposition of Toeplitz algebra.In Chapter1, we have introduced the backgrounds about Toeplitz operator as well as some relative preparations and main works.In Chapter2, we study a single Toeplitz operator Tφ with symbol φ∈L∞(Ω), which is on the general Bergman space Lαp(1<p<+∞), and further conclude that Tφ is compact if and only if its Berezin transform vanishes at the boundary of the annulus Ω.In Chapter3, we study a single Toeplitz operator Tφ with symbol φ∈L∞,1, and further conclude that Tφ is compact if and only if its Berezin transform vanishes at the boundary of the annulus.In Chapter4, we study the Toeplitz operator Tu with symbol u∈C1(M), which is on classical Dirichlet space D2, and further give a canonical decomposition S=TS R for some S=∑i=1mΠj=1nTuij in Toeplitz algebra ξ and some R in the commutator ideal Cξ.