We present two novel results about Hilbert space operators which are
nilpotent of order two. First, we prove that such operators are indestructible
complex symmetric operators, in the sense that tensoring them with any operator
yields a complex symmetric operator. In fact, we prove that this property
characterizes nilpotents of order two among all nonzero bounded operators.
Second, we establish that every nilpotent of order two is unitarily equivalent
to a truncated Toeplitz operator.Comment: 7 pages. To appear in Proceedings of the AM

Bob Lutz

D. Timotin

Dan Timotin

Stephan Ramon Garcia

English

arXiv

CiteSeerX

TWO REMARKS ABOUT NILPOTENT OPERATORS OF ORDER TWO

We present two novel results about Hilbert space operators which are nilpotent of order two. First, we prove that such operators are indestructible complex symmetric operators, in the sense that tensoring them with any operator yields a complex symmetric operator. In fact, we prove that this property characterizes nilpotents of order two among all nonzero bounded operators. Second, we establish that every nilpotent of order two is unitarily equivalent to a truncated Toeplitz operator

Garcia, Stephan Ramon

Lutz, Bob, \u2713

Timotin, D.

Scholarship@Claremont

Claremont CollegesScholarship @ ClaremontPomona Faculty Publications and Research Pomona Faculty Scholarship1-1-2012Two Remarks about Nilpotent Operators of OrderTwoStephan Ramon GarciaPomona CollegeBob Lutz '13Pomona CollegeD. TimotinThis Article - preprint is brought to you for free and open access by the Pomona Faculty Scholarship at Scholarship @ Claremont. It has been acceptedfor inclusion in Pomona Faculty Publications and Research by an authorized administrator of Scholarship @ Claremont. For more information, pleasecontact scholarship@cuc.claremont.edu.Recommended CitationGarcia, Stephan Ramon; Lutz, Bob '13; and Timotin, D., "Two Remarks about Nilpotent Operators of Order Two" (2012). PomonaFaculty Publications and Research. 203.http://scholarship.claremont.edu/pomona_fac_pub/203arXiv:1206.5523v2  [math.FA]  15 Jul 2012TWO REMARKS ABOUT NILPOTENTOPERATORS OF ORDER TWOSTEPHAN RAMON GARCIA, BOB LUTZ, AND DAN TIMOTINAbstract. We present two novel results about Hilbert space operators whichare nilpotent of order two. First, we prove that such operators are indestruc-tible complex symmetric operators, in the sense that tensoring them with anyoperator yields a complex symmetric operator. In fact, we prove that thisproperty characterizes nilpotents of order two among all nonzero bounded op-erators. Second, we establish that every nilpotent of order two is unitarilyequivalent to a truncated Toeplitz operator.1. IntroductionIn the following, H denotes a separable complex Hilbert space and B(H) denotesthe set of all bounded linear operators on H. Recall that an operator T in B(H)is called nilpotent if T n = 0 for some positive integer n. The least such n is calledthe order of nilpotence of T . This note concerns two rather unusual properties ofoperators which are nilpotent of order two.The first result involves complex symmetric operators (see Section 2 for back-ground). It is known that every operator which is nilpotent of order two is acomplex symmetric operator (Lemma 1). However, these operators are complexsymmetric in a much stronger sense, for the tensor product of a nilpotent of ordertwo with an arbitrary operator always yields a complex symmetric operator. Weprove in Section 3 that this property actually characterizes nilpotents of order twoamong all nonzero bounded operators.Our second result concerns truncated Toeplitz operators (precise definitions aregiven in Section 4). To be more specific, we prove that every operator which isnilpotent of order two is unitarily equivalent to a truncated Toeplitz operator havingan analytic symbol. This is relevant to a series of open problems, first arising in [1]and developed further in [9], which, in essence, ask whether an arbitrary complexsymmetric operator is unitarily equivalent to a truncated Toeplitz operator (orpossibly a direct sum of such operators).We close the paper with several open questions suggested by these results.2. Complex symmetryBefore proceeding, let us recall a few basic definitions [2–4]. A conjugation ona complex Hilbert space H is a conjugate-linear, isometric involution. We say thatan operator T in B(H) is complex symmetric if there exists a conjugation C on Hsuch that T = CT ∗C. In this case, we say that T is C-symmetric. The terminologyKey words and phrases. Nilpotent operator, complex symmetric operator, Toeplitz operator,model space, truncated Toeplitz operator, unitary equivalence.The first two authors are partially supported by National Science Foundation Grant DMS-1001614.12 S. R. GARCIA, BOB LUTZ, AND D. TIMOTINreflects the fact that an operator is complex symmetric if and only if it has a self-transpose matrix representation with respect to some orthonormal basis. In fact,any orthonormal basis which is C-real, in the sense that each basis vector is fixedby C, yields such a matrix representation.It is known that if T is nilpotent of order two, then T is a complex symmetricoperator. This was first established directly in [8, Thm. 5], using what we nowrecognize as a somewhat overcomplicated argument. Later on, this result wasobtained as a corollary of the more general fact that every binormal operator iscomplex symmetric [10, Thm. 2, Cor. 4]. A much simpler direct proof is providedbelow. In what follows, we denote unitary equivalence by ∼=.Lemma 1. If T is nilpotent of order two, then T is a complex symmetric operator.Proof. If T in B(H) satisfies T (Tx) = T 2x = 0 for every x in H, it follows thatranT ⊆ kerT = ker |T |. Considering the polar decomposition of T , we see thatT ∼=[0 0A 0]⊕ 0 (1)where A is a positive operator with dense range (the zero direct summand, whichacts on kerT ⊖ ranT , may be absent). Without loss of generality, we may assumethat kerT = ranT . If J is any conjugation which commutes with A (the existenceof such a J follows immediately from the Spectral Theorem), we find that[0 0A 0]︸ ︷︷ ︸T=[0 JJ 0]︸ ︷︷ ︸C[0 A0 0]︸ ︷︷ ︸T∗[0 JJ 0]︸ ︷︷ ︸C,whence T is a complex symmetric operator. For operators on a finite dimensional space, there is a quite explicit proof. Indeed,the positive semidefinite matrix A is unitarily equivalent to a diagonal matrix D =diag(λ1, λ2, . . . , λn) where λ1 ≥ λ2 ≥ · · · ≥ λn ≥ 0, whence[0 0A 0]∼=[0 0D 0]∼=n⊕i=1[0 0λi 0]∼=n⊕i=1λj2[1 ii −1].3. Indestructible complex symmetric operatorsIn the following, H and K denote separable complex Hilbert spaces while A andB are bounded operators on H and K, respectively. Recall that the operator A⊗Bacts on the space H⊗K and satisfies‖A⊗B‖H⊗K = ‖A‖H‖B‖K, (2)the subscripts being suppressed in practice. The following relevant lemma is from [3,Sect. 10], where it is stated without proof.Lemma 2. The tensor product of complex symmetric operators is complex sym-metric.Proof. Suppose that A and B are operators and that C and J are conjugations onH and K, respectively, such that A = CA∗C and B = JB∗J . Let ui and vj denoteC-real and J-real orthonormal bases of H and K, respectively. Define a conjugationC ⊗ J on H ⊗ K by first setting (C ⊗ J)(ui ⊗ vj) = ui ⊗ vj on the orthonormalTWO REMARKS ABOUT NILPOTENT OPERATORS OF ORDER TWO 3basis ui⊗ vj of H⊗K and then extending this to H⊗K by conjugate-linearity andcontinuity. One can then check that A⊗B is (C ⊗ J)-symmetric. On the other hand, it is possible for A ⊗ B to be complex symmetric even ifneither A nor B is complex symmetric. The following lemma provides a simplemethod for constructing such examples.Lemma 3. For each A in B(H) and each conjugation J on H, the operator T =A⊗ JA∗J is complex symmetric.Proof. If Φ in B(H⊗H) is defined first on simple tensors by Φ(x⊗ y) = y⊗ x andthen extended to H ⊗H in the natural way, then C = Φ(J ⊗ J) is a conjugationon H⊗H with respect to which T = CT ∗C. Example 4. Suppose H = ℓ2, A = S (the unilateral shift), and J is entry-by-entrycomplex conjugation on ℓ2. There are many ways to see that S is not a complexsymmetric operator [7, Cor. 7], [3, Prop. 1], [2, Ex. 2.14], [10, Thm. 4]. On theother hand, Lemma 3 implies that S ⊗ S∗ is complex symmetric. In fact,S ⊗ S∗ ∼=∞⊕n=0Jn(0),where Jn(0) denotes a n × n nilpotent Jordan block, which is complex symmetricby [3, Ex. 4]. To see this, note that S ⊗ S∗ is unitarily equivalent to the operator[Tf ](z, w) = z(f(z, w)− f(z, 0)w)on the Hardy space H22 on the bidisk and observe that H22 =⊕∞n=0 Pn where Pndenotes the set of all homogeneous polynomials p(z, w) of degree n. Each subspacePn reduces T and T |Pn∼= Jn(0).Having briefly explored the interplay between tensor products and complex sym-metric operators, we come to the following definition.Definition. An operator A in B(H) is called an indestructible complex symmetricoperator if A⊗B is a complex symmetric operator on H⊗K for all B in B(K).Let us note that an indestructible complex symmetric operator must indeed becomplex symmetric since A ⊗ 1 ∼= A. Clearly, indestructibility is a rather strongproperty. In fact, from the definition alone, it is not immediately clear whetherany nonzero examples exist. As we will see, the nonzero indestructible complexsymmetric operators are precisely those operators which are nilpotent of order two.Theorem 5. T is an indestructible complex symmetric operator if and only if T isnilpotent of order ≤ 2.Proof. If A is nilpotent of order ≤ 2, then A⊗B is also nilpotent of order ≤ 2. ByLemma 1, A⊗B is complex symmetric whence A is indestructible.Before embarking on the remaining implication, let us first remark that if T isC-symmetric, thenw(T, T ∗) = Cw(T ∗, T )C (3)holds for each word w(x, y) in the noncommuting variables x, y. This fact will beuseful in what follows.4 S. R. GARCIA, BOB LUTZ, AND D. TIMOTINNow suppose that A is an indestructible complex symmetric operator. For anyother operator B and any word w(x, y), we obtain‖w(A,A∗)‖‖w(B,B∗)‖ = ‖w(A,A∗)⊗ w(B,B∗)‖= ‖w(A⊗B,A∗ ⊗B∗)‖= ‖w(A∗ ⊗B∗, A⊗B)‖ by (3)= ‖w(A∗, A)‖‖w(B∗, B)‖.Since A is complex symmetric we apply (3) again to obtain‖w(A,A∗)‖‖w(B,B∗)‖ = ‖w(A,A∗)‖‖w(B∗, B)‖. (4)Letting w(x, y) = yx2 andB =0 α 00 0 β0 0 0 ,where α, β are non-negative real numbers, a simple computation reveals that‖w(B,B∗)‖ = α2β, ‖w(B∗, B)‖ = αβ2.If α 6= β, then (4) implies that ‖w(A,A∗)‖ = 0 so that (A∗A)A = 0. ThereforeranA ⊆ kerA∗A = kerA whence A2 = 0, as desired. 4. Unitary equivalence to a truncated Toeplitz operatorThe study of truncated Toeplitz operators has been largely motivated by a semi-nal paper of Sarason [12]. We briefly recall the basic definitions, referring the readerto the recent survey article [5] for a more thorough introduction.In the following, H2 denotes the classical Hardy space on the open unit disk.For each nonconstant inner function u, we consider the corresponding model spaceKu := H2⊖uH2. Letting Pu denote the orthogonal projection from L2 onto Ku, foreach ϕ in L∞ we define the truncated Toeplitz operator Auϕ : Ku → Ku by settingAuϕf = Pu(ϕf).Each such operator is C-symmetric with respect to the conjugation Cf = fzu onKu. We say that Auϕ is an analytic truncated Toeplitz operator if the symbol ϕbelongs to H∞, in which case Auϕ = ϕ(Auz ) by the H∞-functional calculus for Auz .A significant amount of evidence has been accumulated which indicates that trun-cated Toeplitz operators provide concrete models for general complex symmetricoperators [1, 9, 13]. In fact, a surprising array of complex symmetric operators canbe shown to be unitarily equivalent to truncated Toeplitz operators. These resultshave led to several open problems and conjectures [1, Question 5.10], [9, Sect. 7].We refer the reader to [5, Sect. 9] for a thorough discussion of the topic.By Lemma 1, we know that operators which are nilpotent of order two arecomplex symmetric. We now go a step further and prove that every such operatoris unitary equivalent to an analytic truncated Toeplitz operator.Before proceeding, we require a few words about Hankel operators. First let usrecall that the Hankel operator Hϕ : H2 → H2− with symbol ϕ in L∞ is the linearoperator defined by Hϕf = P−(ϕf), where P− denotes the orthogonal projectionfromH2 ontoH2− := L2⊖H2. A detailed treatise on the subject of Hankel operatorsis [11]. We refer the reader there for a complete treatment of the subject.TWO REMARKS ABOUT NILPOTENT OPERATORS OF ORDER TWO 5The first result required is the well-known relationship [11, Ch. 1, eq. (2.9)]Auϕ =MuHuϕ|Ku , (5)where u is an inner function and ϕ belongs toH∞. The next ingredient is [11, Ch. 1,Thm. 2.3].Lemma 6. For ψ in L∞, the following are equivalent:(i) kerHψ is nontrivial,(ii) ranHψ is not dense in H2−,(iii) ψ = uϕ for some inner function u and some ϕ in H∞.Finally, we need the following deep result from [14] (see [11, Ch. 12, Thm. 8.1]):Lemma 7. If A ≥ 0 is an operator on a separable, infinite-dimensional Hilbertspace, then the following are equivalent:(i) A is unitarily equivalent to the modulus of a Hankel operator,(ii) A is unitarily equivalent to the modulus of a self-adjoint Hankel operator,(iii) A is not invertible, and kerA is either trivial or infinite-dimensional.The following general result may be of independent interest.Lemma 8. Any positive operator is unitarily equivalent to the modulus of an ana-lytic truncated Toeplitz operator.Proof. Suppose B ≥ 0 and consider the operator B′ = B ⊕ 0, where 0 acts on aninfinite dimensional Hilbert space. By Lemma 7, B′ is unitarily equivalent to themodulus |Hψ| of some Hankel operator Hψ : H2 → H2−. In light of Lemma 6, itfollows that ψ = uϕ for some inner function u and some ϕ in H∞. We may assumethat u and ϕ are coprime, since a common inner factor of both would cancel in theevaluation of ψ = uϕ.By [11, Ch.1, Thm 2.4], the restrictionHˆ : Ku → H2− ⊖ uH2−of Huϕ to Ku is injective and has dense range. In other words, |Hˆ| is unitarilyequivalent to B|(kerB)⊥ . The operatorW : H2− ⊖ uH2− → Kudefined by W =Mu|H2−⊖uH2−is unitary and satisfies Auϕ =WHˆ by (5). Therefore|Auϕ|∼= |Hˆ| ∼= B|(kerB)⊥ .Now let v be an inner function such that dimKv = dimkerB (e.g., a Blaschkeproduct with dimkerB zeros). Noting thatKuv = Ku ⊕ uKv = vKu ⊕Kv,we see that the matrix ofAuvvϕ : Ku ⊕ uKv → vKu ⊕Kvwith respect to the decompositions above is[vAuϕ 00 0].6 S. R. GARCIA, BOB LUTZ, AND D. TIMOTINSince dimKv = dimkerB, we conclude that|Auvvϕ|∼= |Auϕ| ⊕ 0∼= B|(kerB)⊥ ⊕B|kerB = B. Armed with the preceding lemma, we are ready to prove the following theorem.Theorem 9. If T is nilpotent of order two, then T is unitarily equivalent to ananalytic truncated Toeplitz operator.Proof. It follows from (1) that any nilpotent of order two is unitarily equivalent toan operator of the formN = 0 0 00 0 0B 0 0on H⊕H′⊕H, where H,H′ are Hilbert spaces and B ≥ 0 acts on H. By Lemma 8,we may assume H = Ku and that B = |Auϕ| for some inner function u and ϕ inH∞. Let v be an inner function such that dimKv = dimH′ and let ω : H′ → Kvbe unitary. With respect to the decompositionKu2v = Ku ⊕ uKv ⊕ uvKuwe haveAu2vuvϕ = 0 0 00 0 0uvAuϕ 0 0 .SinceAuϕ is complex symmetric, we can write Auϕ = V |Auϕ| with V unitary [4, Cor. 1].If W denotes the unitary operator(IKu ⊕ ω∗u¯⊕ V ∗u¯v¯) : Ku ⊕ uKv ⊕ uvKu → Ku ⊕H′ ⊕Ku,thenW (Au2vuvϕ)W∗ = N,which proves the theorem. Example 10. If A is a noncompact operator on H, then the operatorT =[0 0A 0]on H ⊕ H is unitarily equivalent to an analytic truncated Toeplitz operator AuϕHowever, since any truncated Toeplitz operator whose symbol is continuous on theunit circle must be of the form normal plus compact [6, Thm. 1, Cor. 2], ϕ cannotbe continuous.5. Open questionsWe conclude this note with some questions suggested by the preceding work.Question 1. Formula (3) may be generalized by considering polynomials p(x, y)in two noncommuting variables x, y. Thenp(T, T ∗) = Cp˜(T ∗, T )Cwhere p˜(x, y) is obtained from p(x, y) by conjugating each coefficient. If T is com-plex symmetric, it follows then that‖p(T, T ∗)‖ = ‖p˜(T ∗, T )‖ (6)TWO REMARKS ABOUT NILPOTENT OPERATORS OF ORDER TWO 7holds for every p(x, y). Does the converse hold? That is, if T in B(H) satisfies (6)for every polynomial p(x, y) in two noncommuting variables x, y, does it follow thatT is a complex symmetric operator?Note that considering only words in T and T ∗ is not sufficient to characterizecomplex symmetric operators. Indeed, if S denotes the unilateral shift, then it iseasy to see that ‖w(S, S∗)‖ = ‖w˜(S∗, S)‖ = 1 for any word w(x, y).The following question stems from the proof of Theorem 9.Question 2. If T is unitarily equivalent to a truncated Toeplitz operator, thendoes the operator T ⊕ 0 have the same property?Although partial results in this direction appear in [13, Section 6], the precedingquestion appears troublesome even in low dimensions.References1. J. A. Cima, S. R. Garcia, W. T. Ross, and W. R. Wogen, Truncated Toeplitz operators:spatial isomorphism, unitary equivalence, and similarity, Indiana U. Math. J. 59 (2010),no. 2, 595–620.2. S. R. Garcia, Conjugation and Clark operators, Recent advances in operator-related functiontheory, Contemp. Math., vol. 393, Amer. Math. Soc., Providence, RI, 2006, pp. 67–111.3. S. R. Garcia and M. Putinar, Complex symmetric operators and applications, Trans. Amer.Math. Soc. 358 (2006), no. 3, 1285–1315 (electronic).4. , Complex symmetric operators and applications. II, Trans. Amer. Math. Soc. 359(2007), no. 8, 3913–3931 (electronic).5. S. R. Garcia and W. T. Ross, Recent progress on truncated Toeplitz operators, Fields InstituteProceedings (to appear), http://arxiv.org/abs/1108.1858.6. S. R. Garcia, W. T. Ross, and W.R. Wogen, C∗-algebras generated by truncated Toeplitzoperators, Oper. Theory Adv. Appl. (to appear), http://arxiv.org/abs/1203.2412 .7. Stephan Ramon Garcia, Means of unitaries, conjugations, and the Friedrichs operator, J.Math. Anal. Appl. 335 (2007), no. 2, 941–947. MR 2345511 (2008i:47070)8. , Aluthge transforms of complex symmetric operators, Integral Equations OperatorTheory 60 (2008), no. 3, 357–367. MR 2392831 (2008m:47052)9. Stephan Ramon Garcia, Daniel E. Poore, and William T. Ross, Unitary equivalence to atruncated Toeplitz operator: analytic symbols, Proc. Amer. Math. Soc. 140 (2012), no. 4,1281–1295. MR 286911210. Stephan Ramon Garcia and Warren R. Wogen, Some new classes of complex symmetricoperators, Trans. Amer. Math. Soc. 362 (2010), no. 11, 6065–6077. MR 2661508 (2011g:47086)11. Vladimir V. Peller, Hankel operators and their applications, Springer Monographs in Mathe-matics, Springer-Verlag, New York, 2003. MR 1949210 (2004e:47040)12. D. Sarason, Algebraic properties of truncated Toeplitz operators, Oper. Matrices 1 (2007),no. 4, 491–526.13. E. Strouse, D. Timotin, and M. Zarrabi, Unitary equivalence to truncated Toeplitz operators,Indiana Univ. Math. J. (to appear).14. S. R. Treil′, An inverse spectral problem for the modulus of the Hankel operator, and balancedrealizations, Algebra i Analiz 2 (1990), no. 2, 158–182. MR 1062268 (91k:47060)Department of Mathematics, Pomona College, Claremont, California, 91711, USAE-mail address: Stephan.Garcia@pomona.eduURL: http://pages.pomona.edu/~sg064747Department of Mathematics, Pomona College, Claremont, California, 91711, USAE-mail address: boblutz13@gmail.comSimion Stoilow Institute of Mathematics of the Romanian Academy, PO Box 1-764,Bucharest 014700, RomaniaE-mail address: Dan.Timotin@imar.ro

Two Remarks about Nilpotent Operators of Order Two

Keck Graduate Institute

Claremont Colleges
Scholarship @ Claremont
Pomona Faculty Publications and Research Pomona Faculty Scholarship
1-1-2012
Two Remarks about Nilpotent Operators of Order
Two
Stephan Ramon Garcia
Pomona College
Bob Lutz '13
Pomona College
D. Timotin
This Article - preprint is brought to you for free and open access by the Pomona Faculty Scholarship at Scholarship @ Claremont. It has been accepted
for inclusion in Pomona Faculty Publications and Research by an authorized administrator of Scholarship @ Claremont. For more information, please
contact scholarship@cuc.claremont.edu.
Recommended Citation
Garcia, Stephan Ramon; Lutz, Bob '13; and Timotin, D., "Two Remarks about Nilpotent Operators of Order Two" (2012). Pomona
Faculty Publications and Research. 203.
http://scholarship.claremont.edu/pomona_fac_pub/203
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TWO REMARKS ABOUT NILPOTENT
OPERATORS OF ORDER TWO
STEPHAN RAMON GARCIA, BOB LUTZ, AND DAN TIMOTIN
Abstract. We present two novel results about Hilbert space operators which
are nilpotent of order two. First, we prove that such operators are indestruc-
tible complex symmetric operators, in the sense that tensoring them with any
operator yields a complex symmetric operator. In fact, we prove that this
property characterizes nilpotents of order two among all nonzero bounded op-
erators. Second, we establish that every nilpotent of order two is unitarily
equivalent to a truncated Toeplitz operator.
1. Introduction
In the following, H denotes a separable complex Hilbert space and B(H) denotes
the set of all bounded linear operators on H. Recall that an operator T in B(H)
is called nilpotent if T n = 0 for some positive integer n. The least such n is called
the order of nilpotence of T . This note concerns two rather unusual properties of
operators which are nilpotent of order two.
The first result involves complex symmetric operators (see Section 2 for back-
ground). It is known that every operator which is nilpotent of order two is a
complex symmetric operator (Lemma 1). However, these operators are complex
symmetric in a much stronger sense, for the tensor product of a nilpotent of order
two with an arbitrary operator always yields a complex symmetric operator. We
prove in Section 3 that this property actually characterizes nilpotents of order two
among all nonzero bounded operators.
Our second result concerns truncated Toeplitz operators (precise definitions are
given in Section 4). To be more specific, we prove that every operator which is
nilpotent of order two is unitarily equivalent to a truncated Toeplitz operator having
an analytic symbol. This is relevant to a series of open problems, first arising in [1]
and developed further in [9], which, in essence, ask whether an arbitrary complex
symmetric operator is unitarily equivalent to a truncated Toeplitz operator (or
possibly a direct sum of such operators).
We close the paper with several open questions suggested by these results.
2. Complex symmetry
Before proceeding, let us recall a few basic definitions [2–4]. A conjugation on
a complex Hilbert space H is a conjugate-linear, isometric involution. We say that
an operator T in B(H) is complex symmetric if there exists a conjugation C on H
such that T = CT ∗C. In this case, we say that T is C-symmetric. The terminology
Key words and phrases. Nilpotent operator, complex symmetric operator, Toeplitz operator,
model space, truncated Toeplitz operator, unitary equivalence.
The first two authors are partially supported by National Science Foundation Grant DMS-1001614.
1
2 S. R. GARCIA, BOB LUTZ, AND D. TIMOTIN
reflects the fact that an operator is complex symmetric if and only if it has a self-
transpose matrix representation with respect to some orthonormal basis. In fact,
any orthonormal basis which is C-real, in the sense that each basis vector is fixed
by C, yields such a matrix representation.
It is known that if T is nilpotent of order two, then T is a complex symmetric
operator. This was first established directly in [8, Thm. 5], using what we now
recognize as a somewhat overcomplicated argument. Later on, this result was
obtained as a corollary of the more general fact that every binormal operator is
complex symmetric [10, Thm. 2, Cor. 4]. A much simpler direct proof is provided
below. In what follows, we denote unitary equivalence by ∼=.
Lemma 1. If T is nilpotent of order two, then T is a complex symmetric operator.
Proof. If T in B(H) satisfies T (Tx) = T 2x = 0 for every x in H, it follows that
ranT ⊆ kerT = ker |T |. Considering the polar decomposition of T , we see that
T ∼=
[
0 0
A 0
]
⊕ 0 (1)
where A is a positive operator with dense range (the zero direct summand, which
acts on kerT ⊖ ranT , may be absent). Without loss of generality, we may assume
that kerT = ranT . If J is any conjugation which commutes with A (the existence
of such a J follows immediately from the Spectral Theorem), we find that[
0 0
A 0
]
︸ ︷︷ ︸
T
=
[
0 J
J 0
]
︸ ︷︷ ︸
C
[
0 A
0 0
]
︸ ︷︷ ︸
T∗
[
0 J
J 0
]
︸ ︷︷ ︸
C
,
whence T is a complex symmetric operator. 
For operators on a finite dimensional space, there is a quite explicit proof. Indeed,
the positive semidefinite matrix A is unitarily equivalent to a diagonal matrix D =
diag(λ1, λ2, . . . , λn) where λ1 ≥ λ2 ≥ · · · ≥ λn ≥ 0, whence[
0 0
A 0
]
∼=
[
0 0
D 0
]
∼=
n⊕
i=1
[
0 0
λi 0
]
∼=
n⊕
i=1
λj
2
[
1 i
i −1
]
.
3. Indestructible complex symmetric operators
In the following, H and K denote separable complex Hilbert spaces while A and
B are bounded operators on H and K, respectively. Recall that the operator A⊗B
acts on the space H⊗K and satisfies
‖A⊗B‖H⊗K = ‖A‖H‖B‖K, (2)
the subscripts being suppressed in practice. The following relevant lemma is from [3,
Sect. 10], where it is stated without proof.
Lemma 2. The tensor product of complex symmetric operators is complex sym-
metric.
Proof. Suppose that A and B are operators and that C and J are conjugations on
H and K, respectively, such that A = CA∗C and B = JB∗J . Let ui and vj denote
C-real and J-real orthonormal bases of H and K, respectively. Define a conjugation
C ⊗ J on H ⊗ K by first setting (C ⊗ J)(ui ⊗ vj) = ui ⊗ vj on the orthonormal
TWO REMARKS ABOUT NILPOTENT OPERATORS OF ORDER TWO 3
basis ui⊗ vj of H⊗K and then extending this to H⊗K by conjugate-linearity and
continuity. One can then check that A⊗B is (C ⊗ J)-symmetric. 
On the other hand, it is possible for A ⊗ B to be complex symmetric even if
neither A nor B is complex symmetric. The following lemma provides a simple
method for constructing such examples.
Lemma 3. For each A in B(H) and each conjugation J on H, the operator T =
A⊗ JA∗J is complex symmetric.
Proof. If Φ in B(H⊗H) is defined first on simple tensors by Φ(x⊗ y) = y⊗ x and
then extended to H ⊗H in the natural way, then C = Φ(J ⊗ J) is a conjugation
on H⊗H with respect to which T = CT ∗C. 
Example 4. Suppose H = ℓ2, A = S (the unilateral shift), and J is entry-by-entry
complex conjugation on ℓ2. There are many ways to see that S is not a complex
symmetric operator [7, Cor. 7], [3, Prop. 1], [2, Ex. 2.14], [10, Thm. 4]. On the
other hand, Lemma 3 implies that S ⊗ S∗ is complex symmetric. In fact,
S ⊗ S∗ ∼=
∞⊕
n=0
Jn(0),
where Jn(0) denotes a n × n nilpotent Jordan block, which is complex symmetric
by [3, Ex. 4]. To see this, note that S ⊗ S∗ is unitarily equivalent to the operator
[Tf ](z, w) = z
(
f(z, w)− f(z, 0)
w
)
on the Hardy space H22 on the bidisk and observe that H
2
2 =
⊕∞
n=0 Pn where Pn
denotes the set of all homogeneous polynomials p(z, w) of degree n. Each subspace
Pn reduces T and T |Pn
∼= Jn(0).
Having briefly explored the interplay between tensor products and complex sym-
metric operators, we come to the following definition.
Definition. An operator A in B(H) is called an indestructible complex symmetric
operator if A⊗B is a complex symmetric operator on H⊗K for all B in B(K).
Let us note that an indestructible complex symmetric operator must indeed be
complex symmetric since A ⊗ 1 ∼= A. Clearly, indestructibility is a rather strong
property. In fact, from the definition alone, it is not immediately clear whether
any nonzero examples exist. As we will see, the nonzero indestructible complex
symmetric operators are precisely those operators which are nilpotent of order two.
Theorem 5. T is an indestructible complex symmetric operator if and only if T is
nilpotent of order ≤ 2.
Proof. If A is nilpotent of order ≤ 2, then A⊗B is also nilpotent of order ≤ 2. By
Lemma 1, A⊗B is complex symmetric whence A is indestructible.
Before embarking on the remaining implication, let us first remark that if T is
C-symmetric, then
w(T, T ∗) = Cw(T ∗, T )C (3)
holds for each word w(x, y) in the noncommuting variables x, y. This fact will be
useful in what follows.
4 S. R. GARCIA, BOB LUTZ, AND D. TIMOTIN
Now suppose that A is an indestructible complex symmetric operator. For any
other operator B and any word w(x, y), we obtain
‖w(A,A∗)‖‖w(B,B∗)‖ = ‖w(A,A∗)⊗ w(B,B∗)‖
= ‖w(A⊗B,A∗ ⊗B∗)‖
= ‖w(A∗ ⊗B∗, A⊗B)‖ by (3)
= ‖w(A∗, A)‖‖w(B∗, B)‖.
Since A is complex symmetric we apply (3) again to obtain
‖w(A,A∗)‖‖w(B,B∗)‖ = ‖w(A,A∗)‖‖w(B∗, B)‖. (4)
Letting w(x, y) = yx2 and
B =

0 α 00 0 β
0 0 0

 ,
where α, β are non-negative real numbers, a simple computation reveals that
‖w(B,B∗)‖ = α2β, ‖w(B∗, B)‖ = αβ2.
If α 6= β, then (4) implies that ‖w(A,A∗)‖ = 0 so that (A∗A)A = 0. Therefore
ranA ⊆ kerA∗A = kerA whence A2 = 0, as desired. 
4. Unitary equivalence to a truncated Toeplitz operator
The study of truncated Toeplitz operators has been largely motivated by a semi-
nal paper of Sarason [12]. We briefly recall the basic definitions, referring the reader
to the recent survey article [5] for a more thorough introduction.
In the following, H2 denotes the classical Hardy space on the open unit disk.
For each nonconstant inner function u, we consider the corresponding model space
Ku := H
2⊖uH2. Letting Pu denote the orthogonal projection from L
2 onto Ku, for
each ϕ in L∞ we define the truncated Toeplitz operator Auϕ : Ku → Ku by setting
Auϕf = Pu(ϕf).
Each such operator is C-symmetric with respect to the conjugation Cf = fzu on
Ku. We say that A
u
ϕ is an analytic truncated Toeplitz operator if the symbol ϕ
belongs to H∞, in which case Auϕ = ϕ(A
u
z ) by the H
∞-functional calculus for Auz .
A significant amount of evidence has been accumulated which indicates that trun-
cated Toeplitz operators provide concrete models for general complex symmetric
operators [1, 9, 13]. In fact, a surprising array of complex symmetric operators can
be shown to be unitarily equivalent to truncated Toeplitz operators. These results
have led to several open problems and conjectures [1, Question 5.10], [9, Sect. 7].
We refer the reader to [5, Sect. 9] for a thorough discussion of the topic.
By Lemma 1, we know that operators which are nilpotent of order two are
complex symmetric. We now go a step further and prove that every such operator
is unitary equivalent to an analytic truncated Toeplitz operator.
Before proceeding, we require a few words about Hankel operators. First let us
recall that the Hankel operator Hϕ : H
2 → H2− with symbol ϕ in L
∞ is the linear
operator defined by Hϕf = P−(ϕf), where P− denotes the orthogonal projection
fromH2 ontoH2− := L
2⊖H2. A detailed treatise on the subject of Hankel operators
is [11]. We refer the reader there for a complete treatment of the subject.
TWO REMARKS ABOUT NILPOTENT OPERATORS OF ORDER TWO 5
The first result required is the well-known relationship [11, Ch. 1, eq. (2.9)]
Auϕ =MuHuϕ|Ku , (5)
where u is an inner function and ϕ belongs toH∞. The next ingredient is [11, Ch. 1,
Thm. 2.3].
Lemma 6. For ψ in L∞, the following are equivalent:
(i) kerHψ is nontrivial,
(ii) ranHψ is not dense in H
2
−,
(iii) ψ = uϕ for some inner function u and some ϕ in H∞.
Finally, we need the following deep result from [14] (see [11, Ch. 12, Thm. 8.1]):
Lemma 7. If A ≥ 0 is an operator on a separable, infinite-dimensional Hilbert
space, then the following are equivalent:
(i) A is unitarily equivalent to the modulus of a Hankel operator,
(ii) A is unitarily equivalent to the modulus of a self-adjoint Hankel operator,
(iii) A is not invertible, and kerA is either trivial or infinite-dimensional.
The following general result may be of independent interest.
Lemma 8. Any positive operator is unitarily equivalent to the modulus of an ana-
lytic truncated Toeplitz operator.
Proof. Suppose B ≥ 0 and consider the operator B′ = B ⊕ 0, where 0 acts on an
infinite dimensional Hilbert space. By Lemma 7, B′ is unitarily equivalent to the
modulus |Hψ| of some Hankel operator Hψ : H
2 → H2−. In light of Lemma 6, it
follows that ψ = uϕ for some inner function u and some ϕ in H∞. We may assume
that u and ϕ are coprime, since a common inner factor of both would cancel in the
evaluation of ψ = uϕ.
By [11, Ch.1, Thm 2.4], the restriction
Hˆ : Ku → H
2
− ⊖ uH
2
−
of Huϕ to Ku is injective and has dense range. In other words, |Hˆ| is unitarily
equivalent to B|(kerB)⊥ . The operator
W : H2− ⊖ uH
2
− → Ku
defined by W =Mu|H2
−
⊖uH2
−
is unitary and satisfies Auϕ =WHˆ by (5). Therefore
|Auϕ|
∼= |Hˆ| ∼= B|(kerB)⊥ .
Now let v be an inner function such that dimKv = dimkerB (e.g., a Blaschke
product with dimkerB zeros). Noting that
Kuv = Ku ⊕ uKv = vKu ⊕Kv,
we see that the matrix of
Auvvϕ : Ku ⊕ uKv → vKu ⊕Kv
with respect to the decompositions above is[
vAuϕ 0
0 0
]
.
6 S. R. GARCIA, BOB LUTZ, AND D. TIMOTIN
Since dimKv = dimkerB, we conclude that
|Auvvϕ|
∼= |Auϕ| ⊕ 0
∼= B|(kerB)⊥ ⊕B|kerB = B. 
Armed with the preceding lemma, we are ready to prove the following theorem.
Theorem 9. If T is nilpotent of order two, then T is unitarily equivalent to an
analytic truncated Toeplitz operator.
Proof. It follows from (1) that any nilpotent of order two is unitarily equivalent to
an operator of the form
N =

 0 0 00 0 0
B 0 0


on H⊕H′⊕H, where H,H′ are Hilbert spaces and B ≥ 0 acts on H. By Lemma 8,
we may assume H = Ku and that B = |A
u
ϕ| for some inner function u and ϕ in
H∞. Let v be an inner function such that dimKv = dimH
′ and let ω : H′ → Kv
be unitary. With respect to the decomposition
Ku2v = Ku ⊕ uKv ⊕ uvKu
we have
Au
2v
uvϕ =

 0 0 00 0 0
uvAuϕ 0 0

 .
SinceAuϕ is complex symmetric, we can write A
u
ϕ = V |A
u
ϕ| with V unitary [4, Cor. 1].
If W denotes the unitary operator
(IKu ⊕ ω
∗u¯⊕ V ∗u¯v¯) : Ku ⊕ uKv ⊕ uvKu → Ku ⊕H
′ ⊕Ku,
then
W (Au
2v
uvϕ)W
∗ = N,
which proves the theorem. 
Example 10. If A is a noncompact operator on H, then the operator
T =
[
0 0
A 0
]
on H ⊕ H is unitarily equivalent to an analytic truncated Toeplitz operator Auϕ
However, since any truncated Toeplitz operator whose symbol is continuous on the
unit circle must be of the form normal plus compact [6, Thm. 1, Cor. 2], ϕ cannot
be continuous.
5. Open questions
We conclude this note with some questions suggested by the preceding work.
Question 1. Formula (3) may be generalized by considering polynomials p(x, y)
in two noncommuting variables x, y. Then
p(T, T ∗) = Cp˜(T ∗, T )C
where p˜(x, y) is obtained from p(x, y) by conjugating each coefficient. If T is com-
plex symmetric, it follows then that
‖p(T, T ∗)‖ = ‖p˜(T ∗, T )‖ (6)
TWO REMARKS ABOUT NILPOTENT OPERATORS OF ORDER TWO 7
holds for every p(x, y). Does the converse hold? That is, if T in B(H) satisfies (6)
for every polynomial p(x, y) in two noncommuting variables x, y, does it follow that
T is a complex symmetric operator?
Note that considering only words in T and T ∗ is not sufficient to characterize
complex symmetric operators. Indeed, if S denotes the unilateral shift, then it is
easy to see that ‖w(S, S∗)‖ = ‖w˜(S∗, S)‖ = 1 for any word w(x, y).
The following question stems from the proof of Theorem 9.
Question 2. If T is unitarily equivalent to a truncated Toeplitz operator, then
does the operator T ⊕ 0 have the same property?
Although partial results in this direction appear in [13, Section 6], the preceding
question appears troublesome even in low dimensions.
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Department of Mathematics, Pomona College, Claremont, California, 91711, USA
E-mail address: Stephan.Garcia@pomona.edu
URL: http://pages.pomona.edu/~sg064747
Department of Mathematics, Pomona College, Claremont, California, 91711, USA
E-mail address: boblutz13@gmail.com
Simion Stoilow Institute of Mathematics of the Romanian Academy, PO Box 1-764,
Bucharest 014700, Romania
E-mail address: Dan.Timotin@imar.ro


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Two remarks about nilpotent operators of order two

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