On the spectrum of n-tuples of p-hyponormal operators

ON THE SPECTRUM OF n -TUPLES OF p-HYPONORMAL OPERATORS by B. P. DUGGAL (Received 27 June, 1996) 1. Introduction. Let B{H) denote the algebra of operators (i.e., bounded linear transformations) on the Hilbert space H.AEB(H) is said to be p-hyponormal (0<p<l), if (AA*Y<{A*Af. (Of course, a l-hyponormal operator is hyponormal.) The p-hyponormal property is monotonic decreasing in p and a /?-hyponormal operator is <7-hyponormal operator for all 0<q <p. Let A have the polar decomposition A = U \A\, where U is a partial isometry and \A\ denotes the (unique) positive square root of A*A. If A has equal defect and nullity, then the partial isometry U may be taken to be unitary. Let 9W(p) denote the class of p-hyponormal operators for which U in A = U \A\ is unitary. $f(/(l/2) operators were introduced by Xia and %tU{p) operators for a general 0<p < 1 were first considered by Aluthge (see [1,14]); fflU(p) operators have since been considered by a number of authors (see [3,4,5,9,10] and the references cited in these papers). Generally speaking, df€U(p) operators have spectral properties similar to those of hyponormal operators. Indeed, let A e S^U(p), (0<p <l/2), have the polar decomposition A = U\A\, and define the SW(p + 1/2) operator A by A = \A\112 U \A\il2. Let /4 = V|/4| with V unitary and A be the hyponormal operator defined by A = |y4|"2V \A\in. Then we have the following result. LEMMA 0. as(A) = as(A), where as denotes either of the following: point spectrum, approximate point spectrum, eigenvalues of finite multiplicity, spectrum, Weyl spectrum, and essential spectrum. Recall that an n-tuple si = (A],A2,.. • ,An) of operators is said to be doubly commuting if A,Aj — AjA, = 0 and AfAj — AjA* = 0, for all X^ia^j^n. Doubly commuting n-tuples si of operators in S€U(p) have been considered by Muneo Cho in [3], where it is shown that a weak Putnam theorem holds for si and that si is jointly normaloid. In this note we study the relationship between the spectral properties of si and s$= (AUA2,... ,An), and prove that cr,(si) = as(si), where crs is either the joint point spectrum or the joint approximate point spectrum or the joint (Taylor) spectrum. This then leads us to: (a) (b) if a{s€) e SV', then A-, is self-adjoint, for all 1 < / < n. We show that the (Cho-Takaguchi) joint Weyl spectrum of si is contained in the (Taylor) spectrum a{s£) of si minus the set of isolated points of a(si) which are joint eigenvalues of finite multiplicity, and that si and si have the same (Harte) essential spectrum. We conclude this note with a result (in the spirit of Dash [8, Corollary 4.6]) on the joint eigenvalues of si in the Calkin algebra. We assume henceforth, without loss of generality, that 0 </? < 1/2. Most of the notation that we use in this note is standard (and usually explained at the first instance of Glasgow Math. J. 40 (1998) 123-131. Downloaded from https://www.cambridge.org/core. IP address: 107.173.28.49, on 06 May 2020 at 16:16:20, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0017089500032419 124 B. P. DUGGAL occurence). The following theorem, the n-tuple version of the Berberian extension theorem, will play an important role in the sequel. THEOREM B. If' si = (Au A2, • • • , An) is an n-tuple of commuting operators on H, then there exists a Hilbert space H°=>H and an isometric ^-isomorphism A/—>A°, ( l < « < n ) , preserving order such that o-n(Aj) = crn(Af) = crp(A°) and an(si) = crK(Ax,A2,... , A,,) = <rK(AuAl...,A0n) = ap(A0uA02,...,A°a) = ap(^). (Here ap(si) and an(s4) denote, respectively, the joint spectrum and the joint approximate point spectrum (defined below) of si.) It is my pleasure to thank Professor Muneo Cho for supplying me with off-prints and preprints of his papers. 2. Results. Throughout the following si = (At,A2,... ,A2) will denote a doubly commuting (i.e., AtAj - AjAj = 0 and A/A* - A*A: - 0, for all 1 < / ¥^j :£ n)n - tuple of XU(p) operators A-, (1 < i <n). Given A< = U, \A,\, define ^,_by A, = |>l,|1Q U, \A,\m; also, letting Aj have the polar decomposition A, = V-t\A-\, define A, by The n-tuples si and si are then defined by si= (AUA2,... (A\,A2,... ,An). ,An) and six = LEMMA 1. s4 is doubly commuting =^>si is doubly commuting^si is doubly commuting. Also, si is doubly commuting ^> [Ah \Aj\] = 0 = [^4,-, \Aj\] = 0, for 1 < i ¥=j < n, where [A, B] denotes the commutator AB - BA of A and B. Proof. Given A, = U, \A{\ and A, = K|/4,|, the doubly commuting hypothesis on si implies that for all 1 < i ¥^j < n. (See [11, Theorems 2 and 4].) Consequently, si is doubly commuting and so for all 1 <i¥^j<n. This implies that si is doubly commuting. The argument above also Hence, also, implies that [AhAj] = [AhAf] = [AhAj] = [AhAf] = 0, for all \<i^j<n. [A,, \Aj\] = [A,, \Aj\] = 0, for all l < » V / < n . In the following we shall denote the Taylor joint spectrum of si by <r(si). (See [13] for the definition of Taylor spectrum of a commuting n-tuple of operators.) We say that A = (Al5 A 2 ,..., An), (A, e C for all l s i < n ) , is in the joint approximate point spectrum o-n(s4) of si if there exists a sequence {xk} of unit vectors in H such that ||(A-A,>*||^0 as fc^°o, for all 1 < / < n ; A = (A,, A2, • . . , kn), A, e C for all 1 <i <n, is in the joint point spectrum ap(si) of si if there exists a non-trivial vector x e H such that 04,-A,)x=0, for all !</<«. Downloaded from https://www.cambridge.org/core. IP address: 107.173.28.49, on 06 May 2020 at 16:16:20, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0017089500032419 n-TUPLES OF p-HYPONORMAL OPERATORS 125 We say that A = (A,, A 2 ,... , A,,) is in the normal point spectrum o-np(sd) of si if there exists a non-trivial vector x e H such that (A,,- A,)x = 0<=>(A,•- A,)*JC = 0, for all 1 < i < n. LEMMA 2. ap{sd) = <rnp(si) = anp{d) = ap(d). Proof. Let A = (A,, A 2 ,.. . , A,,) e o-p{M) and let xeH be such that x^O and (A, - \;)x = 0, for all 1 < i < n. It is easily seen that A | i , | 1 / 2 |A|" 2 = \At\m \A,\mAr, hence for all 1 < / < n. Let /I A where ' " " on the product " I I " denotes that only those \A,\s, (and so also \A,\s), appear in the product for which A, in A,x = \,x does not equal 0. Then y is non-trivial, and Aiy = kiy, for all / = 1 , 2 , . . . , « for which A,^0. If A, = 0, i.e. Atx = 0, then |A-|1/2x = 0. This implies that AjX = 0. Since this in turn implies that |/4,| l/2 x = 0, we conclude that Atx = 0. Since [Ah Aj] = 0 for all 1 s i ^ j £ n , we have that A-,y = 0. Consequently, A e o-p{d) and CTP(^) ^ o- p (^). If, on the other hand, A = (A,, A 2 ,... , A,,) e ap(M), then there is a non-trivial x e / / such that (i4I.-A/)jt = 0 and ( ^ f - A , ) x = 0 for all l < / < n . Since /If |/t,|1/2 |i,| 1 / 2 = for all 1 < i < n. Denning (0 ^ )y by n _ where It' has meaning similar to that above, we have Afy = A,y, for all / = 1 , 2 , . . . , n such that A, ^ 0. Since A, E crp{Aj) implies A, e (TP{A) = crnp(Aj) (see Lemma 0), A,y = A,y for all / = 1,2,... , n such that A, ¥^ 0. Now if A.x - 0, then 0 e o-p(A) = o-p(Ai) and i \Ai\U2x = 0 0&A*x = 0 m f \A,\ x = 0 O A \A,\ll2x=0 (Line 2 follows since 0 e <TP{A,). Line 4 follows because 0 e a-p{A,) = o-,,/;(/4,).) Consequently, Aty = 0 for such an /. Hence o-p{d) c ap(d). Since trp(>!,•) = o-np{A) and a-/;(/4,) = crH/,(/4(-), for all \<i<n, this completes the proof. Downloaded from https://www.cambridge.org/core. IP address: 107.173.28.49, on 06 May 2020 at 16:16:20, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0017089500032419 126 B. P. DUGGAL LEMMA 3. an(si) = ann(s4) = ann{d) = an{d). Proof. Letting A° = (A°u A\,... , A°n) denote the Berberian extension of si (see Theorem B), it follows from Lemma 2 that We are now in a position to prove the equality of the (Taylor) spectra of si and si. THEOREM 1. a(si) = a(si). Proof. Let A = (A,, A 2 ,... , \n) e a(si). Then there exists a partition of {i\,---,LW{ju...,js} {l,2,...,n} and a sequence {xk} of unit vectors in H such that and (Af - \j)xk -»0 (Air-Xir)xk-*0 as /t-^oo, for all l < / - < m and 1 < r < 5. (See [7, Corollary 3.3].) Let si0 denote the Berberian extension (/I?,...,i4? m ,>l°,...,i4?) of si, and let S3 = (/!?„...,/I?,, y4?*,..., A°*). Then (A,v .. . , A,,,,, A/p . . . , Ay) e ap(®). Since (rp(Al) = <rp(/4°) = o-n/,(^i°), for all 1 < r < m , and since /[£ |iy,|1/2 V* \Aj)llzU* = |^y,|1/2 K* \Af2 UfAl it follows (from an argument similar to that used in the proof of Lemma 2) that and A 6 <Tp{d°*) = (7,(^*) £ ff(^*). Hence A e a(d), and Conversely, if A e o-(si), then (from an argument similar to that above) A e ap(si0*). This implies that A e an(si*) c o-(^*), A e o-(^) and a ( ^ ) g (T(S4). Hence o-(^) = cr(^), and the proof is complete. The joint spectral radius r(SP) and the joint operator norm \\9~\\ of an n-tuple ST=(T\,T2,..., Tn) are defined by r i n r{T) = sup||A| = ^ \ 1/2 |A,|2j -i : A = (A,, A 2 ,... , An) e a(ST)j and •xeH, \\x\\= l}. See [6]. The operators si and si being jointly normaloid (see [3, Theorem 9] and [6, Theorem 3.4]), r ( ^ ) = | | ^ | | and r(si) = \\s$\\. Theorem 1 thus implies the following result. Downloaded from https://www.cambridge.org/core. IP address: 107.173.28.49, on 06 May 2020 at 16:16:20, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0017089500032419 n -TUPLES OF p-HYPONORMAL OPERATORS COROLLARY 127 1. | | ^ | | = \\d\\ = \\d\\. That \\A\\ = \\A || for a single operator A e 3W(p) has been proved by M. Fujii et al. in [10]. Given a semi-normal (i.e., hyponormal or co-hyponormal) operator T = X + iY, a well known result of Putnam [12] states that if a real number r E O-(X) (or r + is e a(T), for some real numbers r and s), then there exists a real number 5 such that r + is e a(T) (resp., r e a(X) and s e (T(Y)). This result extends to doubly commuting /r-tuples of hyponormal operators [4]. Does a similar result hold (for A e S^U(p) and) doubly commuting n-tuples in fflU(p)'? The technique of this paper (seemingly) does not lend to a proof of this. We do however have the following analogue for 3%!U(p) operators of a result on n -tuples of doubly commuting hyponormal operators with spectrum in IR". (See [4, Corollary].) COROLLARY 2. / / a{si) £ R", then A, is self-adjoint, for all 1 < i < /z. Proof. Since a(d) = a(d) £ R", A-, is self-adjoint, for all l < / < « , by [4]. Recall that A-, is normal if and only if A-, is normal [9, Corollary 2]; hence At is self-adjoint, for all lsi<«, Following Cho [2], we define the joint Weyl spectrum u^ST) of a commuting n-tuple o"c«(^) = l"l W(y + 3*0; ^ is a n «-tuple of compact operators and (5" + 3if) is a commuting n-tuple}. Let Ooo(^) denote the set of isolated points of o-(ST) which are joint eigen-values of finite multiplicity of ST. It is clear from Theorem 1 that, if A is an isolated point of o-(d), then A is an isolated point of o~(d). The operator d being a doubly commutitive n-tuple of hyponormal operators, an isolated point A of a(d) is a point of ap(d). Hence by Lemma 2 we have the following result. COROLLARY 3. //A is an isolated point of a{M), then A e ap(stf). Recall that if A is p-hyponormal, then aa(A) = a(A) - am(A) by [9] and if ST is a doubly commuting n-tuple of hyponormal operators, then a^ST) £ a(ST) - am(SP) by [2]. T H E O R E M 2. c Proof. Suppose (A,, A 2 , . . . , A,,) e am(d), f n 1 and let N = k e n 2 (A;, — A,)*(y4,• — A,) t. In \ l/= ' Since A E o-p{d) if and only if 0 E ap\ z 04, - A)*04, - A,) I, N is finite dimensional. By Lemma 2, crp(d) - anp(d); hence N reduces d,d() = d | N = (A, | N, A2 \ N,... , A,, \ N) is normal and d\ - d \ N1- = (A, \ NX,A2 \ Nx,. ..,An | N1) is a doubly commuting n-tuple of dtU(p) operators. Let P be the orthogonal projection of H onto N. P is then a compact operator which satisfies [Ah P] = [Af, P] = 0, for all / = 1,2,..., n. The operator _L ' Vn ' 2 _L _L V^ ' ' " ' " Vn Downloaded from https://www.cambridge.org/core. IP address: 107.173.28.49, on 06 May 2020 at 16:16:20, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0017089500032419 128 B. P. DUGGAL is a doubly commuting n-tuple. Let m = (s£ + 9>)\N = ({{A1+^=p\ \N,(A2 + ^ P \ §t and Sf are then doubly commuting n-tuples such that a{s£ + 9) = cr(gft.)Ucr(y). Suppose that A e cr(s4 + $P). Then A £ a{0l) and so A must be an isolated point of cr{Sf). There exists a partition {*,,..., /,„} U {ju ... ,/,} of { 1 , 2 , . . . , n} and a sequence {xk} of unit vectors in /V-1 such that ^0 and { ^ ' ^ ) k But then A e a(si\) and hence (by Corollary 3) A e a p (^,). Thus there exists an x e Nx such that (A, — A,)JC = 0, for all i = l,2,...,n. Since this is a contradiction, we must have A£c REMARKS, (i) the Taylor-Weyl spectrum of ST, <TTW{ST), is defined to be the set of A = (A,,A 2 ,... ,A,,) such that (3T-A) is not Taylor-Weyl (where °T-A is said to be Taylor-Weyl if ST-A is Fredholm and index (9 r -A) = 0). Theorem 2 implies that cr(M)\aTu>(M)^ CTOQ(M). The inclusion a(si)\crru)($$) g o-(X)(^) does not hold (even for hyponormal ,s#). (ii) Given a p-hyponormal operator A, aM(A) = a,o(A) by [9]. Does am d The Warte spectrum a-,,(T) of the commutative n-tuple ST is defined to be an(Sr) = a'(T)l)o-r($), where cr'(^) (respectively, ar(T)) is the set of A = (A,, A 2 ,..., A,,) such that {7; - A,}1£,<,, generates a proper left (resp., right) ideal in B(H). The (Harte) essential spectrum cre{ST) is defined by cre{ST) = cr(a), where a = (aua2,. • • ,an) = K{ST) and n is the canonical homomoprhism of B(H) onto the Calkin algebra B(H)/K(H); K(H) is the algebra of compact operators on H. For a single linear operator, the (Harte) essential spectrum coincides with the essential spectrum; the following extends the conclusion (Te(A) = (Te(A) of Lemma 0 to ae(sd). THEOREM 3. ae(s$) = <re(d). Proof. Suppose A G ac,(d). Then, d being a hyponormal /i-tuple, there exists a sequence {xk} of unit vectors converging weakly to 0 in H such that ||(i,-A,)*xJ|^0 as A:^a>, for all l s i s = n , by [8, Theorem 2.6]. Let {yk} be the sequence defined by ( ) / | | n where '"" on the product Ft denotes that only those \A\ s and \A\ s appear in the product Downloaded from https://www.cambridge.org/core. IP address: 107.173.28.49, on 06 May 2020 at 16:16:20, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0017089500032419 «-TUPLES OF p-HYPONORMAL OPERATORS 1/2 1/2 129 2 for which A,^0. (Notice that if |P,-| **|| or || K| IAI" * J ^ 0 as k-*«>, for some i with 1 < / < « , then |||/4,|l/2jrA.|| and ||^4,-JC,-|| —>0 as &-»<».) Since (xk,h)->0 as A:^oo for all h e H, (yk,h)^0 as k—»°° and y - Ay)*x, «0 as A: for all 1 <y < /?. Thus A e ae(d) and o-e Consider now A e cre($2) = o-'e(s£) U o-re($l). Suppose that A e cr',(^); then there exists a sequence {.v^.} of unit vectors converging weakly to 0 in H such that \\{At - A,)x,|| - » 0 as k —> oo, for all 1 < i < «. Defining the sequence {^A.} by W where IT has a meaning similar to that above , an argument similar to that above shows that {yk} is a sequence of unit vectors converging weakly to 0 in H such that \\(Aj- Aj)yk\\^>0 as fc^oo, for all 1 < / < n . Hence A s a[,(s$). A similar argument shows that if A e (ire{si) then A e a[,(si). Thus £ ae(s£), and the proof is complete. COROLLARY 4. <xe(.s#) = Proof. The argument of the proof of Theorem 3 implies that are(d) c ae(d) = ac(d) = are{d) £ <rre{d). COROLLARY 5. aH{sd) = aL,(s4) U ap(s4*)*. Proof. Let a^si) = {A = (A,, A 2 ,... , A,,): there exists a sequence {xk} of unit vectors in H such that \\(A,•- A,)*JCA. || —^- 0 as &—»°°, for all / = 1 , 2 , . . . , « } denote the joint approximate defect spectrum of siI. Then U U By Lemma 3, an{s^) = a^{si); applying an argument similar to that used in the proof of Lemma 2 to A1-* it is seen that <TS(S$) = as(d). We have )U - an{d) U as(d) = as(d) = aH(d), Downloaded from https://www.cambridge.org/core. IP address: 107.173.28.49, on 06 May 2020 at 16:16:20, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0017089500032419 130 B. P. DUGGAL since si is a hyponormal n-tuple. Also, since (TH{d) = <Ts(d) = <Je(d) U (Tp(d*)* = <Te(d) U <Tp(M*)*, the proof is complete. The /j-tuple (Au A2, ...,An) is said to be essentially doubly commuting (resp., essentially fflU(p)) if the n-tuple (aua2, • • • , an), where a, = n(A;) for all 1 < / < n , and n:B(H)-> B(H)\K(H), is doubly commuting (resp., 9W(p)). We close this note with the following result. THEOREM 4. Suppose (AUA2, • •. ,An) is an n-tuple of essentially doubly commuting have a common reducing subspace essentially 9W(p) operators. Then AuA2,...,An "modulo the compact operators". Proof. The hypotheses imply that a, e dKV(p) for all l < / < « and that the a,s are doubly commuting. Since o-'e(si) n are(d) is not empty (this is consequence of the definition of essential spectrum—see [8, Lemma 4.2]), there exists A = (A,, A 2 , . . . , A,,) e cr'e{sd) n crre(s£) and a non-zero projection q in (the Calkin algebra) B(H)/K(H) such that Since o-p(a,) = crnp(a,), this implies that afq = Xtq (1 < / < « ) . Consequently a,-<7 = (A,<7)* = (flf4)* = <7a, ( l s / < n ) , or, letting n{Q) = q, (AtQ - QA,) is a compact operator, for all !<«'<«. This completes the proof. REFERENCES 1. Ariyadasa Aluthge, On p-hyponormal operators for 0<p < 1, Integral Equations Operator Theory 13 (1990), 307-315. 2. Muneo Ch5, On the joint Weyl spectrum III, Ada Sci. Math. (Szeged) 56 (1992), 365367. 3. Muneo Cho, spectral properties of p-hyponormal operators, Glasgow Math. J. 36 (1994), 117-122. 4. Muneo Cho and A. T. 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Putnam, On the spectra of semi-normal operators, Trans. Amer. Math. Soc. 119 (1965), 509-523. Downloaded from https://www.cambridge.org/core. IP address: 107.173.28.49, on 06 May 2020 at 16:16:20, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0017089500032419 n-TUPLES OF p-HYPONORMAL OPERATORS 131 13. J. L. Taylor, A joint spectrum for several commuting operators, J. Functional Analysis 6 (1970), 172-191. 14. D. Xia, Spectral theory of hyponormal operators (Birkhauser Verlag, Basel, 1983). DEPARTMENT OF MATHEMATICS AND STATISTICS COLLEGE OF SCIENCE, SULTAN QABOOS UNIVERSITY P.O. Box 36, AL-KHOD 123 SULTANATE OF OMAN E-mail: [email protected] Downloaded from https://www.cambridge.org/core. IP address: 107.173.28.49, on 06 May 2020 at 16:16:20, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0017089500032419