11

数理解析研究所講究録
第2073巻 2018年 11-17

ON REFLEXIVITY OF C‐SYMMETRIC OR
SKEW−C‐SYMMETRIC OPERATORSKAMILA KLIŚ‐GARLICKA AND MAREK PTAK
ABSTRACT. We present results concerning reflexivity and hyper‐

reflexivity of a subspace of all C‐symmetric operators from [6] anda subspace of all skew-C‐symmetric operators from [2] with a given
conjugation C . We also give a description of theirs preanihilators.

1. INTRODUCTION

Let \mathcal{H} denote a complex separable Hilbert space with an inner prod‐

uct\rangle and let B(\mathcal{H}) be the Banach algebra of all bounded linear

operators on \mathcal{H}.
Recall that C is a conjugation on \mathcal{H} if C:\mathcal{H}\rightarrow \mathcal{H} is an antilinear,isometric involution, i.e., {Cx, Cy\rangle=\langle y , x) for all x, y\in \mathcal{H} and C^{2}=I.

An operator T in B(\mathcal{H}) is said to be C ‐symmetric if CTC= $\tau$*
Denote by C\{T \in B(\mathcal{H}) : CTC= T^{*}\} the subspace of all C=

symmetric operators.

Operators which are C‐symmetric have beenlately studied by many authors (see [3], [4], [5]). In this class there
are for example Jordan blocs, truncated Toeplitz operators and Hankel

operators. An operator T\in B(\mathcal{H}) is called to be skew-C ‐symmetric
if CTC= -T^{*} Denote by C^{S}\{T \in B(\mathcal{H}) : CTC= -T^{*}\} the=subspace of all skew-C‐symmetric operators. It follows directly fromthe definition that C and C^{s} are weak’ closed. It is worth to note thatany operator T \in B(\mathcal{H}) can be written as a sum of a C‐symmetric

operator and a skew-C‐symmetric operator.

Indeed, T=

A+B,

where A=\displaystyle \frac{1}{2}(T+CT^{*}C) and B=\displaystyle \frac{1}{2}(T-CT^{*}C) .

Recall that the predual to B(\mathcal{H}) is the space of trace class operators
denoted by $\tau$ c with the dual action \{T, f\rangle =tr(Tf) , where T\in B(\mathcal{H})and f\in $\tau$ c . The norm in $\tau$ c is denoted by \Vert \Vert_{1} and called the trace

norm. Denote by F_{k} the set of all operators which have rank at mostk . Rank one operators are usually written as x\otimes y , where x, y\in \mathcal{H},2010 Mathematics Subject Classification. Primary 47\mathrm{A}15 , Secondary 47\mathrm{L}05.
Key words and phrases. \mathrm{s}\mathrm{k}\mathrm{e}\mathrm{w}-C‐symmetry, C‐symmetry, conjugation, reflexiv‐
ity, hyperreflexivity.
The research of the first and the second authors was financed by the Ministry of
Science and Higher Education of the Republic of Poland.

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KAMILA KLIŚ‐GARLICKA AND MAREK PTAK

and (x\otimes y)z\langle z, y\rangle x for z \in \mathcal{H} . Moreover, tr(T(x\otimes y)) =\{Tx, y\rangle.
For a closed subspace S\subset B(\mathcal{H}) denote by s_{\perp} the preanihilator of S
defined by \mathcal{S}_{\perp}= { t\in $\tau$ c:tr(St)=0 for all S\in S }.
Recall that the reflexive closure of a subspace S\subset B(\mathcal{H}) is given by=

Ref S= { T\in B(\mathcal{H}) : Tx\in[Sx] for all x\in \mathcal{H} },hereis the norm‐closure. A subspace S is called reflexive, if \mathcal{S}Ref S and \mathcal{S} is called transitive, if Ref S
B(\mathcal{H}) . A subspace \mathcal{S} \subsetkS^{(k)}
B(\mathcal{H}) is called ‐reflexive if=\{S^{(k)} : S\in S\} is reflexive in B(\mathcal{H}^{(k)}) ,S^{(k)}\mat=S\oplus\cdots\oplusS
where
and hcal{H}^{(k)} =\mathcal{H}\oplus\cdots\oplus \mathcal{H} . Recall after [10, 8]that a weak closed subspace \mathcal{S} is k ‐reflexive if and only if operatorsof rank at most k are linearly dense in s_{\perp} , i.e., S_{1}[S_{\perp}\cap F_{k}] . On
the other hand, transitivity means that there are no rank-1 operatorsin the preanihilator of S , i.e., s_{\perp}\cap F_{1} =\{0\}.
=

=

*

=

The definition of k‐hyperreflexivity was introduced in [1, 7] andis a stronger property than k‐reflexivity, which means that each k‐
hyperreflexive subspace is k ‐reflexive. A subspace S is called k‐hyper‐
reflexive if there is a constant c>0 such that(1)

dist (T, \displaystyle \mathcal{S})\leq c\cdot\sup\{|tr(Tt)| : t\in F_{k}\mathrm{n}s_{\perp}, \Vert t\Vert_{1} \leq 1\},for all T\in B(\mathcal{H}) . Note thàt dist (T, S)=\displaystyle \inf\{\Vert T-S\Vert: S\in \mathcal{S}\} and thesupremum on the right hand side of (1) we denote by $\alpha$_{k}(T, S) . The
smallest constant for which the inequality (1) is satisfied is called thek ‐hyperreflexivity constant and is denoted $\kappa$_{k}(\mathcal{S}) . If k=1 , the letter kwill be omitted.

In this paper we present results concerning reflexivity and hyper‐

reflexivity of subspaces C and C^{s} proved in [6] and [2]. It is shown thatthe subspace of all C‐syninietric operators is transitive (hence far frombeing reflexive) and 2‐reflexive or even 2‐hyperreflexive. It means thatthe preanihilator of C does not contain any rank‐one operators andrank‐two operators are linearly dense in the preanihilator. Moreover,
we describe all rank‐two operators in this preanihilator. However, thesubspace of all skew-C‐symmetric operators have much better proper‐

ties: it is reflexive (so very far from being transitive) and hyperreflexive.
2. PREANIHILATOR

Let \mathcal{H} be a complex separable Hilbert space with an antilinear in‐volution C . Now we will present results describing the structure ofpreanihilator of the subspace C . First theorem says that there are no
rank-1 operators in t,he preanihilator.

Theorem 2.1 (Theorem 2.1 [6]). Let C be the set of C ‐symmetric
operators. The subspace C is transitive.

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REFLEXIVITY OF C AND C^{s}

The next theorem gives a full description of rank-2 operators in c_{\perp}.

Theorem 2.2 (Theorem 3.1 [6]). Let C be the set of all C ‐symmetric
operators. Then

F_{2}\cap C_{\perp}=\{h\otimes g-Cg\otimes Ch:h, g\in \mathcal{H}\}.
Let now consider some examples of conjugations given in [3] in thecontext of Theorem 2.2.

Example 2.3. A natural example of a conjugation in l^{2}(\mathbb{N}) is given by

C(z_{0}, z_{1}, z_{2}, \ldots)=(\overline{z}_{0},\overline{z}_{1},\overline{z}_{2}, \ldots).

In this case

c_{\perp}\cap F_{2}=\{h\otimes g-\overline{g}\otimes\overline{h} : h, g\in l^{2}(\mathrm{N})\}.
Example 2.4. Consider the classical Hardy space H^{2} and take a non‐H^{2}\ominus $\alpha$ H^{2} For f \in K_{ $\alpha$}^{2}
coristarrt inner function a . Denote by K_{ $\alpha$}^{2}=

and h\in H^{2} the formula

C_{ $\alpha$}f= $\alpha$\overline{zf}defines a conjugation C=C_{ $\alpha$} on K_{ $\alpha$}^{2} . Then

C_{1}\cap F_{2}=\{h\otimes g- $\alpha$\overline{zg}\otimes $\alpha$\overline{zh} : h, g\in K_{ $\alpha$}^{2}\}.
Example 2.5. Let $\rho$ be a bounded, positive continuous weight on theinterval [−1, 1], symmetric with respect to the midpoint of the interval:$\varrho$(t)= $\rho$(-t) for t\in[0 , 1 ] . Then the formulaCf(t)=\overline{f(-t)}
defines a conjugation on L^{2}([-1,1], $\rho$ dt) . In this case

c_{\perp}\cap F_{2}=\{h(\cdot)\otimes g-\overline{g(-(\cdot))}\otimes\overline{h(-(\cdot))}:h, g\in L^{2}([-1,1], $\rho$ dt

Example 2.6. Consider the isometric antilinear operator

C(z_{1}, z_{2})=(\overline{z}_{2}\prime,\overline{z}_{1})on \mathb {C}^{2}Then

C_{\perp}\cap F_{2}=\{(h_{1}, h_{2})\otimes(g_{1}, g_{2})-(\overline{g}_{2}, \overline{g}_{1})\otimes(\overline{h}_{2},\overline{h}_{1}):(h_{1}, h_{2}), (g_{1}, g_{2})\in \mathbb{C}^{2}\}.
Now we will consider the preanihilator of the subspace of all skew‐
C‐synmietric operators.

Lemma 2.7. Let C be a conjugation in a complex Hilbert space \mathcal{H} andh, g\in \mathcal{H} . Then

(1) C(h\otimes g)C=Ch\otimes Cg,(2) h\otimes g-Cg\otimes Ch\in C^{s}14
KAMILA KLIS‐GARLICKA AND MAREK PTAK

In [3, Lemma 2] it was shown that
C\cap F_{1}=\{ $\alpha$\cdot h\otimes Ch:h\in \mathcal{H}, $\alpha$\in \mathbb{C}\}.
The next proposition shows that it is also a description of the rank‐one
operators in the preanihilator of C^{s}

Proposition 2.8 (Proposition 2.2 [2]). Let C be a conjugation in acomplex Hilbert space \mathcal{H} . Then

C_{\perp}^{s}\cap F_{1}=C\cap F_{1}=\{ $\alpha$\cdot h\otimes Ch:h\in \mathcal{H}, $\alpha$\in \mathbb{C}\}.
Lemma 2.9. Let C be a conjugation in a complex Hilbert space \mathcal{H}.
Then

C_{\perp}^{s}\cap F_{2}\supset\{h\otimes g+Cg\otimes Ch:h, g\in \mathcal{H}\}.
The following examples illustrate the result presented in Proposition
2.8.

Example 2.10. Note that for different conjugations we obtain different
subspaces. Let C_{1} (x_{1}, x_{2}, x3)(\overline{x}_{3}, \overline{x}_{2}, \overline{x}_{1}) be a conjugation on \mathb {C}^{3}
=

Then

and

\mathcal{C}_{1}^{s}=\{ left(\begin{ar y}{l } a&b&0\ c&0&-b\ 0&-c&-a \end{ar y}\right):a,b c\in\mathb {C}\C_{1}=\{ left(\begin{ar ay}{l } a&b&*\ c&*&b\ *&c&a \end{ar ay}\right):a,b c\in\mathb {C}\Rank‐one operators in C_{1} and in (C_{1}^{s})_{\perp} are of the form $\alpha$(x_{1}, x_{2}, x_{3})\otimes(\overline{x}_{3},\overline{x}_{2},\overline{x}_{1}) for $\alpha$\in \mathbb{C}.

If we now consider another conjugation C_{2} (x_{1}, x_{2}, x3)
on \mathb {C}^{3} , thenand

C_{2}^{s}=\{.\left(\begin{ar ay}{l } a&0&b\ 0&-a&c\ -c&-b&0 \end{ar ay}\right):a,b c\in\mathb {C}\,C_{2}=\{ left(\begin{ar ay}{l } a&*&b\ *&a&c\ c&b&* \end{ar ay}\right):a,b c\in\mathb {C}\=(\overline{x}_{2},\overline{x}_{1}, \overline{x}_{3})Similarly, rank‐one operators in C_{2} and also in (C_{2}^{s})_{\perp} are of the form$\alpha$(x_{1}, x_{2}, x_{3})\otimes(\overline{x}_{2},\overline{x}_{1},\overline{x}_{3}).

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REFLEXIVITY OF C AND C^{s}

Example 2.11. Let C be a conjugation in \mathcal{H} . Consider the conjuga‐tion \tilde{C}=\left(\begin{ar y}{l 0&C\ C&0 \end{ar y}\right) in\mathcal{H}\oplus \mathcal{H} (see [9]).

An operator T\in B(\mathcal{H}\oplus \mathcal{H})is skew-\tilde{C}‐symmetric, if and only if T=

\left(\begin{ar ay}{l} A&B\ D&-CA^{*}C \end{ar ay}\right), whereA, B, D \in B(\mathcal{H}) and B, D are skew-C‐symmetric. Moreover, rank‐
one operators in \tilde{C}_{\perp}^{s} are of the form $\alpha$(f\oplus g)\otimes(Cg\oplus Cf) for f, g\in \mathcal{H}and $\alpha$\in \mathbb{C}.

The following example gives a description of skew-C‐symmetric op‐
erators in a case of model space K_{ $\alpha$}^{2} equipped with the conjugation C_{ $\alpha$}
defined in Example 2.4.
Example 2.12. Let H^{2} be the Hardy space, and let $\alpha$ be a non‐constant inner function. As in Example 2.4 consider the conjugation$\alpha$\overline{zh} on the space K_{ $\alpha$}^{2}H^{2}\ominus $\alpha$ H^{2} By S_{ $\alpha$} and S_{ $\alpha$}^{*} denote
C_{ $\alpha$}hthe compressions of the unilateral shift S and the backward shift S^{*}==

to K_{ $\alpha$}^{2} , respectively. Recall after [11] that the kernel functions in K_{ $\alpha$}^{2}

for $\lambda$\in \mathbb{C} are projections of appropriate kernel functions k_{ $\lambda$} onto K_{ $\alpha$}^{2},
namely k_{ $\lambda$}^{ $\alpha$}=k_{ $\lambda$}-\overline{ $\alpha$( $\lambda$)} $\alpha$ k_{ $\lambda$} . Denote by \tilde{k}_{ $\lambda$}^{ $\alpha$}=C_{ $\alpha$}k_{ $\lambda$}^{ $\alpha$} . Since S_{ $\alpha$} and S_{ $\alpha$}^{*} are
C_{ $\alpha$} ‐symmetric (see [3]), for a skew-C_{ $\alpha$} ‐symmetric operator A\in B(K_{ $\alpha$}^{2})

we have(2)\langle AS_{ $\alpha$}^{n}k_{ $\lambda$}^{ $\alpha$} , (S_{ $\alpha$}^{*})^{m}\tilde{k}_{ $\lambda$}^{ $\alpha$}\}=\langle C_{ $\alpha$}(S_{ $\alpha$}^{*})^{m}\tilde{k}_{ $\lambda$}^{ $\alpha$}, C_{ $\alpha$}AS_{ $\alpha$}^{n}k_{ $\lambda$}^{ $\alpha$}\rangle=-\langle S_{ $\alpha$}^{m}C_{ $\alpha$}\tilde{k}_{ $\lambda$}^{ $\alpha$}, A^{*}C_{ $\alpha$}S_{ $\alpha$}^{n}k_{ $\lambda$}^{ $\alpha$}\rangle=-\langle AS_{ $\alpha$}^{m}k_{ $\lambda$}^{ $\alpha$}, (S_{ $\alpha$}^{*})^{n}\tilde{k}_{ $\lambda$}^{ $\alpha$}\},
for all n, m\in \mathbb{N} . Note that if n=m , then(3)\{AS_{ $\alpha$}^{n}k_{ $\lambda$}^{ $\alpha$}, (S_{ $\alpha$}^{*})^{n}\tilde{k}_{ $\lambda$}^{ $\alpha$}\}=0.

z^{k}, k > 1.
In particular, we may consider the special case $\alpha$
Then the equality (3) implies that a skew-C_{ $\alpha$} ‐symmetric operator A\in
B(K_{z^{k}}^{2}) has the matrix representation in the canonical basis with 0 on
the diagonal orthogonal to the main diagonal. Indeed, let A\in B(K_{z^{k}}^{2})
have the matrix (a_{ij})_{i,j=0,\ldots k-1}\prime.
with respect to the canonical basis. Note
=

that

have

C_{z^{k}}f=z^{k-1}\overline{f}, k_{0}^{z^{k} =\cdot 1, \tilde{k}_{0}^{z^{k} =z^{k-1}

Hence for 0\leq n\leq k-1 we

0=\langle AS_{ $\alpha$}^{n}1, (S_{ $\alpha$}^{*})^{n}z^{k-1}\rangle\backslash =\langle Az^{n}, z^{k-n-1}\rangle=a_{n\prime k-n-1}.
Moreover, from the equality (2) we can obtain that\{Az^{n}, z^{k-m-1}\}=-\langle Az^{m}, z^{k-n-1}\rangle,
which implies that a_{n\prime k-m-1}=-a_{m,k-n-1} for 0\leq m, n\leq k-1.

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KAMILA KLIŚ‐GARLICKA AND MAREK PTAK
3. REFLEXIVITY

In this section we present results concerning reflexivity of the space of
all C‐synimetric operators and the subspace of all skew-C‐symmetric
operators.

Theorem 3.1 (Theorem 4.1 [6]). Let \mathcal{H} be a complex separable Hilbert
space with an antilinear involution C. The subspace C \subset B(\mathcal{H}) of all
C ‐symmetric operators is 2‐reflexive.

In the case of the space of all skew-C‐syninietric operators we canobtain a stronger result.

Theorem 3.2 (Theorem 3.1 [2]). Let C be a conjugation in a complex
Hilbert space \mathcal{H} . The subspace C^{s} of all skew-C ‐symmetric operators
on \mathcal{H} is reflexive.

Recall that, a single operator T \in B(\mathcal{H}) is called reflexive if the
weakly closed algebra generated by T and the identity is reflexive.

In [9] authors characterized normal skew symmetric operators and by[12] we know that every normal operator is reflexive. Hence one may

wonder, if all skew-C‐symmetric operators are reflexive. The following
simple example shows that it is not true.
Example 3.3. Consider the space \mathb {C}^{2} and a conjugation C(x, y)(\overline{x}, y\left(\begin{ar y}{l 0&\mathrm{l}\ -1&0 \end{ar y}\right)\mathcal{A}(T)\left(\begin{ar y}{l a&b\ -b\mathcal{A}(T)_{\perp}\cap&a \end{ar y}\right)\mathcal{A}(T)_{\perp}=\{\left(\begin{ar ay}{l } t & s\ s & -t \end{ar ay}\right) : t, s\in \mathb {C}\}F_{1}=\{0\}Note that operator T=

to see, thatis skew-C‐symmetric. The

generated by T consists of operators of the

weakly closed algebraform=. Hence. It is easy

, which implies that T is not reflexive.

4. HYPERREFLEXIVITY

Hyperreflexivity is a stronger property than reflexivity. Here we
present results concerning hyperreflexivity of the subspaces C and C^{s}
Since C is transitive, it cannot be hyperreflexive. However, we canprove the following:

Theorem 4.1 (Theorem 4.2 [6]). Let \mathcal{H} be a complex separable Hilbert

space and let C be a conjugation on \mathcal{H} . The sub_{\mathcal{S}}paceC\subset B(\mathcal{H}) of all

C ‐symmetric operators is 2‐hyperreflexive with constant 1.

The subspace C^{s} is reflexive. It can be proved that it also has thestronger property—hyperreflexivity.

17
REFLEXIVITY OF C AND C^{s}

Theorem 4.2 (Theorem 4. 1 [2] ). Let C be a conjugation in a com‐plex Hilbert space \mathcal{H} . Then the subspace C^{s} \subseteq B(\mathcal{H}) of all skew-Csymmetric operators is hyperreflexive with the constant $\kappa$(C^{s}) \leq 3 and2‐hyperreflexive with $\kappa$_{2}(C^{s})=1.
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[8] J. Kraus and D: R. Larson, Reflexivity and distance formulae, Proc. London
Math. Soc. 53 (1986), 340‐356.
:[9] C. G. Li and S. Zhu: Skew symmetric normal operators, Proc. Amer. Math.
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[10] A.I. Loginov and V.S. Shul’man: Hereditary and intermediate reflexivity ofW^{*}- algebras, Izv. Akad. Nauk. SSSR, 39 (1975), 1260−1273; Math. USSR‐Izv.
9 (1975), 1189−1201[11] D. Sarason, Algebraic properties of truncated Toeplitz operators, Operators andMatrices 1 no. 4 (2007), 491‐526.
[12] —, Invariant subspaces and unstarred operator algebras, Pacific J.
Math. 17 no. 3 (1966): 511‐517.
KAMILA KLiś‐GARLici\langle A, INSTITUTE OF MATHEMATICS, UNIVERSITY OF AGRI‐CULTURE,BALICKA 253c,30‐l98 KRAKOW,, POLAND
E‐mail address:

rmklisQcyfronet. pl

MAREK PTAK, INSTITUTE OF MATHEMATICS, UNIVERSITY OF AGRICULTURE,
BALICKA 253\mathrm{c} , 30‐198 KRAKOW, POLAND, AND INSTITUTE OF MATHEMATICS,

PEDAGOGICAL UNIVERSITY, \mathrm{U}\mathrm{L} . PODCHORAZYCH 2, 30‐084 KRAKóW, POLAND
E‐mail address: rmptak@cyfronet.pl