Weighted Shift Operators

Let H be a complex separable Hilbert space and B(H) is the Banach algebra of all bounded linear operators on H. An operator S ∈ B(H) is an unilateral weighted shift if there is an orthonormal basis {e_n,n ∈ Z} and a sequence of scalars {a_n} such that Se_n = a_ne_n+1 for all n ∈ Z. Say that S is a weighted shift with weighted sequence {a_n}. Bilateral weighted shifts are denned analogously. An operator S ∈ B(H) is a Bilateral weighted shift with weighted sequence {a_n,n∈ Z} if there is an orthonormal basis {e_n,n ∈ Z} such that Se_n = a_ne_n+1 for all n.let {W_n}_n=1^∞ be a uniformly bounded sequence of operators on H, the operator S on (l|^)²(H) defined by S : {x₀, x₁, …} = {0, W₁x₀, W₂X₁, …} is called an operator weighted shift on (l|^)²(H) with the weighted sequence {W_n}_n=1^∞, denoted by S {W_n}.The fundamental reference for weighted shift is Shields . Weighted shifts constitute a testing ground for operator theory, whenever an operator theorist thinks of a new concept, the first reaction, after seeing what it says in the finite dimentional case, is to examine this for weighted shifts . Are there examples of weighted shifts having this property? The result of such an inquiry will often indicate whether the top has sufficient merit to warrant further investigation.The operator weighted shift was first studied by Lambert. It is a natural generalization of the scalar weighted shift operator and owns many properties similar to those of the scalar one.The thesis consists of there chapters.In the first chapter, We introduce the basic concepts, the background ofIn the first chapter, We introduce the basic concepts, the background of the problems.In the second chapter ,we give a sufficieent and necessary condition for a hyponormal operator weighted shift to be a near subnormal operator .Therem 2.11 Let Wn be one-to-one and commuting positive operators in B{H) such that 0 < Wx < W2 < W3 < ? â–  ? and defineS{xo,xux2,---} = {W1x1,W2x2,WaX3,- ? ?}â–  Then operator S is near subnormal if and only if(1) x e Ker(W2 - W^) =? Wnx 6 Ker(WB2+1 -(2) R((Wn+i W&) = R((W*+1 - W&) ,where Wo = 0.Therem 2.12 Let Wn be a one-to-one and commuting positive operators in B{H) such that ? < W_x < Wo < Wi < â–  â–  â–  and defineS{- â–  â–  , x-u [xo},xu -..} = {..., W_iX₂, [Wox_!], WlXo, â– â– â– } Then operator S is near subnormal if and only if(1) x € Ker(Wn2 - W^) =* Wnx e(2) lIn the third chapter, we give a sufficient and necessary condition for an operator which is the restriction of multiplication operator to its invariant subspaces to be a weighted shift operator .Therem 3.8 Let jjl is a positive Borel measure on closed unitdisc and Mz is a multiplication operator on L2(/i),Then the statements (1) and (2) below are equivalent:(1) There is a positive borel measure v defined on the closed unit interval [0,1] , f(z) is a positive measurable function on B, ï¿¡2 is a nonempty subset of D such that Xnd/J, = f(z)dv x dA(0), where dA(0) is the normalized lebesgue measure on [0, 2tt].(2) There is a nontrivial invariant subspace M. such that Mz\m is a weighted shift operator.Therein 3.9 Let /i is a positive Borel measure on closed unit disc and Mz is a multiplication operator on ï¿¡2(/x), Then the statements (1) and (2) below are equivalent:(1) There is a positive Borel measure v defined on the closed unit interval [0,1], f(z) is a positive measurable function on D, ï¿¡1 is a nonempty measure subset of D such that Xfid/J = f(z)du x dA(#), where dA(#) is the normalized lebesgue measure on [0, 2tt] with r13 € L1^) for all real /? .(2) There is a nontrivial invariant subspace M such that Mz\m is a weighted shift operator.