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 (1917-2006) 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. 2000 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 0 0 0 terrced mtrix M = 1 1 0 2 2 2 is bounded liner opertor on 1 Received 19 Februry, 2007 Accepted for publiction (in revised form) 4 December, 2007 5
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 1 0 0 1 1 0 is no, we consider the cse where M = C = 2 2 1 1 1 3 3 3, 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 1 0 0 subsequence; we will regrd M = 2 2 0 3 3 3 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
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 1 2 1 2 1 3 M (M )* = 1 2 2 2 2 2 2 3 1 3 2 2 3 3 3 2 = (U*M){(UU*)(M*U)} nd tht 2 1 2 2 1 2 3 1 3 2 U*{(MM*)U} = 1 2 3 2 2 3 2 3 2 1 3 3 2 3 4 3 2 = (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 + 0 = 0 for ll f in l 2. This completes the proof of prt ().
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
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), 125-137. [2] G. Leibowitz, Rhly Mtrices, J. Mth. Anl. Appl. 128 (1987), 272-286. [3] H. C. Rhly, Jr., Terrced Mtrices, Bull. Lond. Mth. Soc. 21 (1989), 399-406. [4] H. C. Rhly, Jr., Posinorml Opertors, J. Mth. Soc. Jpn 46 (4) (1994), 587-605. 1081 Buckley Drive Jckson, Mississippi 39206 E-mil: rhly@lumni.virgini.edu, rhly@member.ms.org