Stability And Spectral Invariant Of Convolution Dominated Operators

In this paper,we use the finite propagation ideas to discuss stability and spectral invariant of convolution-dominated operators.Firstly, for A∈(?),||A||(?)e=∑κ∈Zdsupi-j=k |a(i,j)|<∞,satisfying (?)c∈lp,1≤p≤∞, where|c|=(|c(j)|j∈Zd),l1(zd)(?)a=(a(κ))κ∈Zd=(supj-j'=κ|a(j,j')κ∈Zd,we discuss itslp-stability at infinity,i.e.(?)C1,C2>0,for (?)k∈灭(lp(Zd),there will exist a constant R:=R(K)>0 so that inequality for(?)ξ∈lp(zd)with supp(ξ)∩B(O,R):φholds.Secondly on countable discrete group G we try to research the spectral invariant of Hl,d∞(G)=∩s≥0∞Hl,ds.S,d(G)and the completeness of Hl,ds(G)for S large enough,where Hl,d∞(G)is a smooth subalgebra of the uniform Roe algebraCu*(G)=Cu,alg*(G)B(l2(G,de)) for(?)T∈Cu,alg*(G),T haS finite propagation and for (?)T∈Cu*(G),it is l2-convolution dominated,where l is the length function on the countable discrete group G, for (?)T∈Cu,alg(G),its norm is defined by for(?)ψ∈Hl,ds(G),its norm is defined by for(?)T∈Cu,alg*(G),T=[t(.x,y)](x,y)∈(G×G),there will exist a constant R>0 such that for (?)(x,y)∈(G×G) with dl(x,y)> R, t(x,y)=0.