Several Kinds Of Weakly Hyponormality And Criteria For Positively Quadratically Hyponormality Weighted Shifts

LetH be a separable infinite dimensional complex Hilbert space, L(H) be the algebra of all bounded linear operators on H,α,α(x) denote weight sequence, Wα, Wα(x) denote unilateral weighted shifts.Let T∈L(H), we call T hyponormal if T*T≥TT*.Let T∈L(H), we call T subnormal if T=N|H,where N is a normal operator on an invariant subspaces of K (?) H.Let T∈L(H), we call T weak hyponormal if is hyponormal, especially, we call T quadratically hyponormal when k=2.An n-tuple T=(T1,…T1) of operators on H is joint hyponormal if is positive on H(?)…(?)H.Let T∈L(H) is 2-hyponormal if T=(T, T2) is joint hyponormalLetα:α0,α1,…is a positive weight sequence,Wαis a positively quadratically hy-ponormal, if (?)n, i≥0,0≤i< (n+1), c(n, i)≥0, where c(n, i) be the McLaurin coefficients of the polynomial dn. R.Curto proved the following theorem in 1990:Let x>0,Wαis an unilateral weighted shifts,its weight sequence isα0=x,αn= then:(1)Wαis subnormal(2)Wαis 2-hyponormalMore generally,there exists a sequence of positive numbers such that:(a)λk>λk+1 for all k≥1(c)Wαis k-hyponormal(?)01).(3)Wαis quadratically hyponormalThen we can obtain the following theorem:Let x>0,Wαis an unilateral weighted shifts,its weight sequence isα0=x,αn= then:(1)Wαis 2-hyponormal but not subnormal(2)Wαis quadratically hyponormal but not subnormal(3)Wαis quadratically hyponormal but not 2-hyponormalThat is:subnormality(?) 2-hyponormality,2-hyponormality(?)quadratically hyponor-mality,but the converse is not true.Young Bin Choi,Jin Kyu and Woo Young Lee proved the following theorem in 2000:Let x>0,Wαis an unilateral weighted shifts,its weight sequence isα0=x^1/2,αn= then:(a)Wαis postively quadratically hyponormal(b)00, Wαis an unilateral weighted shifts, its weight sequence isαo=x^1/2,αn= then Wa is quadratically hyponormal but not positively quadratically hyponor-mal forThat is:postively quadratically hyponormal(?) quadratically hyponormal, but the con-verse is not true.Moreover, we obtained the following theorem:Let x>0, Wαis an unilateral weighted shifts, its weight sequence isα0=x^1/2,an-then: therefore the equation c(n,3)= 0 have two roots, one is 0 and the other is(d) (?)n≥4, let xn is the positive root of equation c(n,3)= 0, then therefore the equation c(n, k)=0 have two roots, one is 0 and the other(f)(?)n≥4, if x≥(28)/(57), then (?)n>k≥3, c(n,k)≤0.(g)(?)n≥k≥4, let xn(k) is the positive root of equation c(n, k)= 0, then:(h) If c(n,k)≥0, then (?)n>k≥4, c(n+1,k)≥0 (?)n> k≥5, c(n,k-1)≥0.Ju Young Bae,Il Bong Jung and George R. Exner proved the following theorems in 2002: Let Wαis 2-hyponormal, then Wa positively quadratically hyponormal.Let Wa be a weighted shift with property B(k) for some k≥2, then Wαis positively quadratically hyponormal if and only if n=1,2,…, k-1, i=1,2,…, k, c(n+i-1,i)≥0.Let Wa be a weighted shift withα0=α1, if Wαwith property B(3), then Wαis positively quadratically hyponormal if and only if c(3,2)≥0 and c(4,3)≥0.thus we can obtain:Subnormality(?) 2-hyponormality,2-hyponormality(?) positively quadratically hy-ponormality, positively quadratically hyponormality(?) quadratically hyponormality,but the converse is not true.Let Wa be a weighted shift with property B(k) for some k≥2, then Wa is positively quadratically hyponormal if and only if n= 1,2,…, k-1, i= 2,…, k, c(n+i-1, i)≥0.Let Wa be a hyponormal weighted shift with property B(3), then Wαis positively quadratically hyponormal if and only if c(3,2)≥0 and c(4,3)≥0.Let Wa be a hyponormal weighted shift with property B(4), u4v3=w3, then Wa is positively quadratically hyponormal if and only if c(4,3)≥0.At last, we give three examples to show the usefulness of the criteria for positively quadratically hyponormal.