A Note on Heredity for Terraced Matrices 1

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1 Generl Mthemtics Vol. 16, No. 1 (2008), 5-9 A Note on Heredity for Terrced Mtrices 1 H. Crwford Rhly, Jr. In Memory of Myrt Nylor Rhly ( ) Abstrct A terrced mtrix M is lower tringulr infinite mtrix with constnt row segments. In this pper it is seen tht when M is bounded liner opertor on l 2, hyponormlity, compctness, nd noncompctness re inherited by the immedite offspring of M. It is lso shown tht the Cesàro mtrix cnnot be the immedite offspring of nother hyponorml terrced mtrix Mthemticl Subject Clssifiction: 47B99 Key words: Cesàro mtrix, terrced mtrix, hyponorml opertor, compct opertor 1 Introduction Assume tht { n } is sequence of complex numbers such tht the ssocited terrced mtrix M = is bounded liner opertor on 1 Received 19 Februry, 2007 Accepted for publiction (in revised form) 4 December,

2 6 H. Crwford Rhly, Jr. l 2 ; these mtrices hve been studied in [2] nd [3]. We recll tht M is sid to be hyponorml on l 2 if [M,M]f,f = (M M MM )f,f 0 for ll f in l 2. It seems nturl to sk whether hyponormlity is inherited by the terrced mtrix rising from ny subsequence { nk }. To see tht the nswer is no, we consider the cse where M = C = , the Cesàro mtrix. In [4, Corollry 5.1] it is seen tht the terrced mtrix ssocited with the subsequence { 1 : n = 0, 1, 2,.} is not hyponorml, lthough 2n+1 the Cesàro mtrix itself is known to be hyponorml opertor on l 2 (see [1]). Consequently, we turn our ttention to more modest result nd consider hereditry properties of the terrced mtrix rising from one specil subsequence; we will regrd M = s the immedite offspring of M, for M is itself the terrced mtrix tht results from removing the first row nd the first column from M. Note tht M = U MU where U is the unilterl shift. 2 Min Result Theorem 2.1. () M inherits from M the property of hyponormlity. (b) M is compct if nd only if M is compct. Proof. () We must show tht [(M ),M ] (M ) M M (M ) 0. Criticl to the proof is the fct tht (M )* M = U*{(M*M)U}, which cn be verified by computing tht both sides of the eqution re equl to the

3 A Note on Heredity for Terrced Mtrices 7 b 1 b 2 b 3 b reverse-l-shped mtrix 2 b 2 b 3 b 3 b 3 b 3 where b n = k 2 ; lso, it k=n cn be verified tht M (M )* = = (U*M){(UU*)(M*U)} nd tht U*{(MM*)U} = = (U*M){I(M*U)}. Consequently, we hve [(M )*,M ] = (M )*M - M (M )* = U*{(M*M)U} - (U*M){(UU*)(M*U)} = U*{(M*M)U} - U*{(MM*)U} + U*{(MM*)U} - (U*M){(UU*)(M*U)} = U*{(M*M)U} - U*{(MM*)U} + (U*M){I(M*U)} - (U*M){(UU*)(M*U)} = U*{[M*,M]U} + (M*U)*{(I - UU*)(M*U)}. Since M is hyponorml (by hypothesis) nd I UU 0, we find tht [(M )*, M ] f,f = = [M*,M] Uf,Uf + ((I UU*)(M*U) f, (M*U) f = 0 for ll f in l 2. This completes the proof of prt ().

4 8 H. Crwford Rhly, Jr. (b) We prove only one direction. Suppose M is compct. It follows tht UM U* is lso compct. Note tht M UM U* hs nonzero entries only in the first column; these entries re precisely the terms of the sequence { n }. Since M is bounded, we must hve n 2 = Me 0 2 <, where e 0 belongs to the stndrd orthonorml bsis for l 2 ; consequently, M UM U is Hilbert-Schmidt opertor on l 2 nd is therefore compct. Thus M = UM U* + (M UM U*) is compct, since it is the sum of two compct opertors. Corollry 2.1. Assume M is the terrced mtrix obtined by removing the first k rows nd the first k columns from M, for some fixed positive integer k > 1. () M inherits from M the property of hyponormlity. (b) M is compct if nd only if M is compct. n=0 3 Other Results We note tht normlity (occurring when M commutes with M*) nd qusinormlity (occurring when M commutes with M*M) re lso inherited properties for terrced mtrices, but those turn out to be trivilities. The proofs re left to the reder. Theorem 3.1. () If M is norml, then n = 0 for ll n 1 nd M = 0. (b) If M is qusinorml, then n = 0 for ll n 1 nd M = 0. In closing, we consider question bout the most fmous terrced mtrix, the Cesàro mtrix C. Is C the immedite offspring of some other hyponorml terrced mtrix; tht is, does there exist hyponorml terrced mtrix A such tht C = A = U*AU? The mtrix A would hve to be generted by { n } with 0 yet to be determined nd n = 1 for n 1. Then n L = lim (n + 1) n + 1 n = lim = 1. From [3, Theorems 2.5 nd 2.6] we n + n + n

5 A Note on Heredity for Terrced Mtrices 9 conclude tht the spectrum is σ(a) = {λ : λ 1 1} { 0 } nd tht A cnnot be hyponorml since n 2 1 = n = π2 2 6 > 1 = L2. Thus we n=1 n=1 see tht nonhyponormlity is not inherited by the immedite offspring of terrced mtrix. References [1] A. Brown, P. R. Hlmos, nd A. L. Shields, Cesàro Opertors, Act Sci. Mth. Szeged 26 (1965), [2] G. Leibowitz, Rhly Mtrices, J. Mth. Anl. Appl. 128 (1987), [3] H. C. Rhly, Jr., Terrced Mtrices, Bull. Lond. Mth. Soc. 21 (1989), [4] H. C. Rhly, Jr., Posinorml Opertors, J. Mth. Soc. Jpn 46 (4) (1994), Buckley Drive Jckson, Mississippi E-mil: rhly@lumni.virgini.edu, rhly@member.ms.org