数理解析研究所講究録
第 1996 巻 2016 年 23-33

23

On $m$-complex symmetric operators
Muneo Cho, Eungil Ko, and Ji Eun Lee

Abstract
In this paper, we provide several spectral and local spectral properties of-complex symmetric operators. Moreover, we study properties of nilpotent
perturbations of $m$-complex symmetric operators. Finally, we discuss the
structures of $m$-complex symmetric operators.
$m$Introduction

1

be theThe results in this paper will be appeared in other journals. Let
algebra of all bounded linear operators on a separable complex Hilbert space
which satisfies
conjugation on is an antilinear operator$C$ ,$C^{2}=I$. For any conjugation there is $an/$orthonormal
for alland(see [12] for more details).
such that $Ce_{n}=e_{n}$ for all
for
basis
Note that $(CTC)^{k}=CT^{k}C$ and $(CTC)^{*}=CT^{*}C$ for every positive integer
is said to be complex symmetric if there
and $\Vert C||=1$ . An operator$C$$T=CT^{*}C$. In this case, we say thaton such thatexists a conjugationis complex symmetric with conjugation $C$ . This terminology is due to the factthat is a complex symmetric operator if and only if it is unitarily equivalent toa symmetric matrix with complex entries, regarded as an operator acting on an-space of the appropriate dimension (see [12]). The class of complex symmetric
operators includes all normal operators, Hankel matrices, finite Toeplitz matrices,
all truncated Toeplitz operators, and some Volterra integration operators. We refer
the reader to $[12]-[15J$ for more details.
$\mathcal{L}(\mathcal{H})$$\mathcal{H}.$$\langle y,$$\langle Cx,$

$C:\mathcal{H}arrow \mathcal{H}$$\mathcal{H}$$A$$Cy\rangle=$$x,$ $y\in \mathcal{H}$$x\rangle$$\{e_{n}\}_{n=0}^{\infty}$$n$$\mathcal{H}$$k,$$T\in \mathcal{L}(\mathcal{H}\rangle$$T$$\mathcal{H}$$T$$t^{2}$

Mathematics Subject Classification; Primary $47Al1$ , Secondary $47B25.$
Keywords; -Complex symmetric operator; Decomposable; Single-valued extension property.
This work was supported by Basic Science Research Program through the National Research
Foundation of $Korea\langle NRF$ ) grant funded by the Korea government MEST) $(2009-0093125)$ . The
third author was supported by Basic Science Research Program through the National Refunded by the Ministry of Education, Science and Technolsearch Foundation of Korea$0_{2010}$$($$(NRF\rangle$$ogy(2012RlAlA3006841)$ .

24
In 1970, J. W. Helton [16] initiated the study of operators
satisfy an identity of the form;$\sum_{j=0}^{m}(-1)^{m-j}(\begin{array}{l}mj\end{array})T^{*j}T^{m-j}=0$$T\in \mathcal{L}(\mathcal{H})$

which

.

(1)

In light of complex symmetric operators, using the identity (1), we define mcomplex symmetric operators as follows; an operator
is said to be an$m$ -complex symmetric operator if there exists some conjugation $C$ such that$T\in \mathcal{L}(\mathcal{H})$$\sum_{j=0}^{m}(-1)^{m-j}(\begin{array}{l}mj\end{array})T^{*j}CT^{m-j}C=0$

for some positive integer $m$ . In this case, we say thatwith conjugation $C$ . Set$T$

is $m$-complex symmetric$\Delta_{m}(T):=\sum_{j=0}^{m}(-1)^{m-j}(\begin{array}{l}mj\end{array})T^{*j}CT^{m-j}C$. Then$T$isan -complex symmetric operator with conjugation if and only if $\Delta_{m}(T)=0.$
An operator
is said to be a strict -complex symmetric operator if
is an -complex symmetric operator but it is not an $(m-1)$ -complex symmetric
$C$

$m$

$T\in \mathcal{L}(\mathcal{H})$

$T$

$m$

$m$

operator.

Note that$T^{*}\triangle_{m}(T)-\triangle_{m}(T)(CTC)=\triangle_{m+1}(T)$.

(2)

Hence, if is $m$ -complex symmetric with conjugation $C$ , thenis -complex
symmetric with conjugation
for all $n\geq m$ . It is obvious that a 1-complex
symmetric operator is complex symmetric.
$T$

$T$

$n$

$C$

2
If

Preliminaries, we write

,
,
,
,
,
,
,
,
, and
for the spectrum, the surjective spectrum, the compression spectrum, the point spectrum, the approximate point spectrum, the essential spectrum,
the left essential spectrum, the right essential spectrum, Browder spectrum, andWeyl spectrum of $T$ , respectively.
An operator
is said to have the single-valued extension property (orSVEP) if for every open subset $G$ of and any -valued analytic function $f$ on$G$such that $(T-\lambda)f(\lambda)\equiv 0$ on $G$ , we have $f(\lambda)\equiv 0$ on $G$ . For an operator$T\in \mathcal{L}(\mathcal{H})$$\sigma(T)$$\sigma_{su}(T)$$\Gamma(T)$$\sigma_{p}(T)$$\sigma_{ap}(T)$$\sigma_{e}(T)$$\sigma_{le}(T)$$\sigma_{re}(T)$$\sigma_{w}(T)$$\sigma_{b}(T)$$T\in \mathcal{L}(\mathcal{H})$$\mathbb{C}$$\mathcal{H}$, the local resolvent set $p_{T}(x)$ of at is definedand for a vector
as the union of every open subset of on which there is an analytic function:
such that $(T-\lambda)f(\lambda)\equiv x$ on . The local spectrum of at isgiven by. We define the local spectral subspace of an operator
by $H_{T}(F)=\{x\in \mathcal{H} : \sigma_{T}(x)\subset F\}$ for a subset $F$ of . An operator$T\in \mathcal{L}(\mathcal{H})$$T$$x\in \mathcal{H}$

$G$

$f$

$Garrow \mathcal{H}$

$G$

$T$

$\sigma_{T}(x)=\mathbb{C}\backslash \rho_{T}(x)$$T\in \mathcal{L}(\mathcal{H})$$x$$\mathbb{C}$$\mathbb{C}$

$x$

25
if for every open subset of andis said to have Bishop’s propertyof -valued analytic functions on such thatevery sequence
converges uniformly to in norm on compact subsets of , we get thatconverges uniformly to in norm on compact subsets of . An operator
is said to be decomposable if for every open cover $\{U, V\}$ of there are -invariant
subspacesand such that$G$$(\beta$ $)$$\tau\in \mathcal{L}(\mathcal{H})$$\{f_{n}\}$$G$$\mathcal{H}$$(T-\lambda)f_{n}(\lambda)$

$G$

$0$

$f_{n}(\lambda)$

$G$

$0$

$T\in \mathcal{L}(\mathcal{H})$$T$$\mathbb{C}$$\mathcal{X}$$\mathbb{C}$$\mathcal{Y}$$\mathcal{H}=\mathcal{X}+\mathcal{Y},$$\sigma(T|_{\mathcal{X}})\subset\overline{U}$, and$\sigma(T|_{\mathcal{Y}})\subseteq\overline{V}.$It is well-known that
Decomposable$\Rightarrow Bishop$’s property$(\beta)\Rightarrow$

SVEP.

In general, the converse implications do not hold (see [20] for more details).

3

ExamplesIn this section, we consider several examples of $m$-complex symmetric operatorswith conjugation . It is well-known that if is nilpotent of order 2, then iscomplex symmetric by [11, Theorem 5]. But if is nilpotent of order with $k>2,$
then may not be complex symmetric.
$T$

$T$

$C$

$k$

$T$

$T$

and let be a conjugation on . If is nilpotent
Example 3.1 Letof order $k>2$ and $\tau*\neq CTC$ , then is $a(2k-1)$ -complex symmetric operator
with conjugation $C.$
$T\in \mathcal{L}(\mathcal{H})$$C$$\mathcal{H}$

$T$

$T$

is a self-adjoint operator
Example 3.2 Let be a conjugation on . If
$Q^{k}=0$$Q^{*}\neq
CQC$
for some $k>2$ , then an
andand $R=CRC$ where $RQ=QR,$
operator $R+Q$ is $(2k-1)$ -compJex symmetric with conjugation $C.$$C$

$R\in \mathcal{L}(\mathcal{H})$$\mathcal{H}$

Example 3.3 Let $C$ be a conjugation given by$T=(\begin{array}{lll}0 1 00 0 20 0 0\end{array})$on$\mathbb{C}^{3}$, then $T^{3}=0$ and$T$$C(z_{1}, z_{2}, z_{3})=(\overline{z_{1}},\overline{z_{2}},\overline{z_{3}})$on$\mathbb{C}^{3}$. If

is a not complex symmetric operator

by [15]. Hexlce is a 5-complex symmetric operator with conjugation $C$ . However,since $T^{3}=0$ and $T^{2}\neq 0$ , it follows that$T$$\sum_{j=0}^{4}(-1)^{4-j}(\begin{array}{l}4j\end{array})T^{*j}CT^{4-j}C=6T^{*2}CT^{2}C=(\begin{array}{lll}0 0 00 0 00 0 24\end{array}) \neq 0.$

So it is not a 4-complex symmetric operator.

26

4

$m$

-complex symmetric operators

In this section, we provide spectral properties of $m$-complex symmetric operators.
Recall that two vectors and are said to be -orthogonal if
$x$

$C$

$y$

$\langle Cx,$$y\rangle=0.$

Theorem 4.1 Let
be an $m$ -complex symmetric operator with conjugation C. Then the following statements hold.
(i) If is an eigenvalue of , then is an eigenvalue of(ii) $Eigenvector\mathcal{S}$ of corresponding to distinct eigenvalues are -orthogonal.
(iii) If
, then.
(iv) Let. If
sequencesaresuch thatof unitand $\lim_{narrow\infty}(T-\mu)y_{n}=0$ , then$T\in \mathcal{L}(\mathcal{H})$$T$$\lambda$$\overline{\lambda}$$\tau*.$$T$$C$$\lambda\in\sigma_{ap}(T)$$\lambda\neq\mu$$\overline{\lambda}\in\sigma_{ap}(T^{*})$$\{x_{n}\},$$\{y_{n}\}$$vector\mathcal{S}$$\lambda)x_{n}=0$$\lim_{narrow\infty}\langle Cx_{n},$$\lim_{narrow\infty}(T-$$y_{n}\rangle=0.$

An operator
is said to be isoloid if for any,
eigenvalue of , where isodenotes the set of all isolated points of, we denote $D^{*}=\{\overline{z}:z\in D\}.$$T\in \mathcal{L}(\mathcal{H})$$\lambda\in iso\sigma(T)$$T$$\sigma(T)$$\lambda\in \mathbb{C}$$\sigma(T)$

is an
. For

$D\subset \mathbb{C}$Corollary 4.2 Let
be $m$ -complex symmetric with conjugation C. If
is isoloid, thenis also isoloid.
$T\in \mathcal{L}(\mathcal{H})$$T$$\tau*$

Theorem 4.3 Ifjugationsuch thatwith conjugation $C.$$\{T_{k}\}$$C$

is a sequence of $m$ -complex symmetric operators with con, then is also $m$ -complex symmetric$T$$\lim_{karrow\infty}\Vert T_{k}-T\Vert=0$

We provide equivalent statements for an $m$-complex symmetric operator.
Proposition 4.4 Let
be invertible and let
be a conjugation on
Then the following sertions hold.
(i) If $T^{*j}CT^{m-j}C=CT^{m-j}CT^{*j}$ for $j=0$ , 1,
, then is $m$ -complex$CT^{*-1}C$
metric with conjugation
is $m$ -complex symmetric with
if and only if$C.$conjugation(ii) is $m$ -complex symmetric with conjugation if and only if
is $m$ -complex
symmetric with conjugation $C.$$C$

$T\in \mathcal{L}(\mathcal{H})$$\mathcal{H}.$$a\mathcal{S}$$\cdots$$\rangle m$$T$$\mathcal{S}ym-$

$C$

$T$

$C$

$T^{-1}$

Theorem 4.5 If
is an $m$ -complex symmetric operator with conjugation, thenis also $m$ -complex symmetric with conjugation $C$ for some positive$n.$integer$T\in \mathcal{L}(\mathcal{H})$$C$$T^{n}$

Corollary 4.6 Let$T\in \mathcal{L}(\mathcal{H})$$\lim_{narrow\infty}\Vert T^{n}x\Vert^{\frac{1}{n}}=0$, thenbe $m$ -complex symmetric with conjugation C.

$\lim_{narrow\infty}\Vert T^{*mn}Cx\Vert^{\frac{1}{n}}=0.$

If

27
Example 4.7 Let $C$ be a conjugation given by

If$T=(\begin{array}{lll}0 l 00 0 20 0 0\end{array})$$(\begin{array}{lll}0 0 00 0 02 0 0\end{array})$. Henceon$T^{2}$$\mathbb{C}^{3}$, then$C(z_{1}, z_{2}, z_{3})=(\overline{z_{3}},\overline{z_{2}},\overline{z_{1}})$on$\mathbb{C}^{3}.$and $T^{*2}=CT^{2}C=$$\tau*\neq CTC=(\begin{array}{lll}0 0 02 0 00 1 0\end{array})$

is a 1-complex symmetric operator but$T$

is not a 1-complex

symmetric operator with conjugation $C.$

We next study the local spectral properties of $m$-complex symmetric operators.
Theorem 4.8 Let
be an $m$ -complex symmetric operator with conjugation C. Then
has the property
if and only if is decomposabie.
$T\in \mathcal{L}(\mathcal{H})$$\tau*$$T$$(\beta$ $)$An operator
is quasiaﬃnity if it has trivial kernel and dense range
andis quasiaﬃne transform of if there is a quasiaﬃnity $X$ such that
$XS=TX$ . Two operators and are quasisimilar if there are quasiaﬃnities $X$
and such that $XS=TX$ and $SY=YT$ . A closed subspaceis invariant, and hyperinvariant for if it is invariant for every operator infor if$\{T\}^{f}=\{S\in \mathcal{L}(\mathcal{H}) : T8=ST\}$ of $T.$
$X\in \mathcal{L}(\mathcal{H})$$T$$S\in \mathcal{L}(\mathcal{H})$

$T$

$S$

$Y$

$\mathcal{M}\subseteq \mathcal{H}$

$T$

$T$

$T\mathcal{M}\subseteq \mathcal{M}$Corollary 4.9 Let
be $m$ -complex symmetric operators. Then the followinghold.
(i) Ifis hyponormal, i.e., $TT^{*}\geq T^{*}T$ , then is decomposable.
(ii) If
has the property
andhas nonempty interior, then has a non$T\in \mathcal{L}(\mathcal{H})$$statement_{\mathcal{S}}$$T$$\tau*$$\tau*$$T$$\sigma(T)$$(\beta$ $)$trivial invariant(iii) If
is not singleton andnontrivial hyperinvariant subspace.
$sub_{\mathcal{S}}pace.$$\sigma(T)$$S\in \mathcal{L}(\mathcal{H})$is quasisimilar to $T$ , then

$S$

has a

Theorem 4.10 Let
be an $m$ -complex symmetric operator with conjuga$\tau*has$tion C. If
the single-vaiued extension property, then has the single-vatued
extension property. Moreover, in this. Furall$CH_{T}(F)\subseteq
H_{T^{x}}(F^{*})$$F^{*}:=\{\overline{z}:z\inF\}$
thermore,
where
for any set in$T\in \mathcal{L}(\mathcal{H})$$T$$case_{f}\sigma_{T^{*}}(x)\subseteq\sigma_{T}(Cx)^{*}for$$x\in \mathcal{H}$$F$$\mathbb{C}.$

Assume that has the single-valued extension property. If there exists a constant such that for every
we have
with
$T$

$k$

$x,$ $y\in \mathcal{H}$$\sigma_{T}(x)\cap\sigma_{T}(y)=\emptyset$$\Vert x\Vert\leq k\Vert x+y\Vert$

where is independent of and , we say that an operator
boundedness condition (B) .
$k$

$x$

$y$

$T$

satisfies Dunford’s

28
Corollary 4.11 Let
be an $m$ -complex symmetric operator with conjugation C. If
has the single-valued extension property and the Dunfor boundedness condition (B) , then also has the Dunford’s boundedness condition (B) .
$T\in \mathcal{L}(\mathcal{H})$$d^{f}s$$\tau*$$T$

Next, we investigate properties of nilpotent perturbations of $m$-complex symmetric operators.
, let $C$ be a conjugation on , and let $N$ be
Proposition 4.12 Let
nilpotent of order $n(n>2)$ with $NT=TN$ . Then the following statements hold.
(i) Assume that is strict $m$ -complex symmetric with the conjugation $form>1.$
If commutes with $CT^{*}C$ and $CN^{*}C$ , then $T+N$ is $a(2n+m-2)$ -complex
symmetric operator.
(ii) Suppose thatis a complex symmetric operator with a conjugation C. If
commutes with $CN^{*}C$ , then an operator $T+N$ is $a(2n-1)$ -complex ymmetric
operator.
$\mathcal{H}$$T\in \mathcal{L}(\mathcal{H})$

$C$

$T$

$T$

$T$

$T$

$\mathcal{S}$is $m$ -complex symmetric with the conjugation andTheorem 4.13 If$N$ is a nilpotent operatorof order with $TN=NT$ , then the following statements
equivalent:are(i) is decomposable.
(ii)
has the property ( ) .
(iii) $T+N$ is decomposable.
(iv) $\tau*+N^{*}$ has the property ( ) .
$C$$T\in \mathcal{L}(\mathcal{H})$

$n$

$T$

$\tau*$$\beta$$\beta$Corollary 4.14 If
is complex symmetric with the conjugation $C$ and$N$ is a nilpotent operatorof order with $TN=NT$ , then the following statements
are equivalent:(i)
is decomposable.
(ii)
has the property ( ) .
(iii)
has the property ( ) .
(iv) $T+N$ is $decompo\mathcal{S}able.$(v) $\tau*+N^{*}$ has the property ( ) .
(vi) $T+N$ has the property ( ) .
$T\in \mathcal{L}(\mathcal{H})$

$n$

$T$

$\tau*$$T$$\beta$$\beta$$\beta$$\beta$Let us recall that we say that Weyl’s theorem holds for$\sigma(T)\backslash \sigma_{w}(T)=\pi_{00}(T)$$T\in \mathcal{L}(\mathcal{H})$if,denotes thewhere $\pi_{00}(T)=\{\lambda\in iso\sigma(T) : 0<dimker(T-\lambda)<\infty\}$ andset of all isolated points of . We say that Browder’s theorem holds for. As some applications of Theorem 4.13, we get the following
if$iso\Delta$$\triangle$$\sigma_{b}(T)=\sigma_{w}(T)$corollary.

$T\in \mathcal{L}(\mathcal{H})$29
Corollary 4.15 Let $R=T+N\in \mathcal{L}(\mathcal{H})$ where is $m$ -complex symmetric and$N^{n}=0$ with $TN=NT$ .
If has the property ( ) , then the following assertions
hold.
(i)
and
have the property
and the single-valued extension property.
(ii) If
has nonempty interior, then has a nontrivial invariant subspace.
(iii) $H_{R}(F)$ is a hyperinvariant subspace for $R.$(iv) If is any function analytic on a neighborhood of, then both Weyl’sand Browder’s theorems hold for $f(R)$ and $\sigma_{w}(f(R))=\sigma_{b}(f(R))=f(\sigma_{w}(R))=$.
$T$$\tau*$$R$$R^{*}$$\beta$$(\beta$ $)$$R$$\sigma(R)$$\sigma(R)$$f$$f(\sigma_{b}(R))$

Lemma 4.16 If
tions hold.
(i)

is an $m$ -complex symmetric operator, then the following rela-$T$$\sigma_{p}(T)\subseteq\sigma_{p}(T^{*})^{*},$$\sigma_{ap}(T)\subseteq\sigma_{ap}(T^{*})^{*},$$\Gamma(T^{*})^{*}\underline{\subseteq}\Gamma(T)_{f}\sigma_{su}(T^{*})^{*}\subseteq\sigma_{su}(T)$$\sigma(T)=\sigma_{ap}(T^{*})^{*}=\sigma_{su}(T)$(ii)(iii) If$\sigma_{le}(T)\subseteq\sigma_{le}(T^{*})^{*},$$\tau*$$\sigma_{re}(T^{*})^{*}\underline{\subseteq}\sigma_{re}(T)$

, and

, and

.

$\sigma_{e}(T)\underline{\subseteq}\sigma_{re}(T)$.

has the single-valued extension property, then$\sigma(T)=\sigma_{ap}(T)=\sigma_{ap}(T^{*})^{*}=\sigma(T^{*})^{*}$

Proposition 4.17 Let $R=T+N$ be an operator inwith the same hypotheses as in Theorem 4.13. Then the following properties hold.
(i)
, and
(ii)
and
$\mathcal{L}(\mathcal{H})$$\sigma_{p}(R)\subseteq\sigma_{p}(T^{*})^{*}\cup\{0\},$$\sigma_{le}(R)\subseteq\sigma_{ie}(T)$5$\Gamma(R^{*})^{*}\subseteq\Gamma(T)\cup\{0\}$$\sigma_{ap}(R)\underline{\subseteq}\sigma_{ap}(T^{*})^{*}\cup\{0\}.$$\sigma_{re}(R^{*})^{*}\subseteq\sigma_{re}(T^{*})^{*}.$

Structure of $m$-complex symmetric operators

In this section, we focus on structures of $m$-complex symmetric operators. For$0<p\leq 1$ , an operator
is said to be -hyponormal if $(T^{*}T)^{p}\geq(TT^{*})^{p}.$We call an operator
skew complex symmetric if there exists a conjugation
on such that $T=-CT^{*}C.$$T\in \mathcal{L}(\mathcal{H})$$p$$T\in \mathcal{L}(\mathcal{H})$$C$$\mathcal{H}$andbe a conjugation on . Then
Theorem 5.1 Let be an operator on
the following statements hold.
(i) If $m$ is even, thenis complex symmetric with the conjugation C. In thiscase, ifis -hyponormal, then it is normal.
odd, then(ii) Ifis skew complex symmetric with the conjugation $C.$
In this case, if $\Delta_{m}(T)=0$ andis $p-hyponormal_{f}$ then $T^{*}\Delta_{m-1}(T)=$$T$$\mathcal{H}$$\triangle_{m}(T)$$\Delta_{m}(T)$$mi_{\mathcal{S}}$$p$$\Delta_{m}(T)$$\triangle_{m-1}(T)$$C$$\mathcal{H}$

30
and

$\triangle_{m-1}(T)CTC$$\triangle_{m-1}(T)$is normal.

(iii) Let
$K_{m}(T):= \bigcap_{n\geq m}ker(\triangle_{n}(T))$

If $K_{1}(T)\neq\{O\}$ andinvariant subspace for$K_{m}(T)\neq \mathcal{H}$.

, then the subspace $C(K_{m}(T))$ is a nontrivial$T.$and $C$ be a conjugation on . Then
Corollary 5.2 Let
be an operator on
the following statements hold.
.
(i) If $m$ isthen$m$(ii) If isthen$m$is even.
(iii) If is odd andhas finite rank , then the rank ofthen $C(K_{1}(T))$ has at least two distinct(iv) If $K_{1}(T)\neq\{O\}$ andelements of$ker(\Delta_{j}(T))$$m-1$ . If
is a strict(v) Put $F_{n}(T)$ $:=$
for $n=1$ , 2,$T$$even_{f}$$odd_{J}$$\mathcal{H}$$\mathcal{H}$$\sigma(\triangle_{m}(T))=\sigma_{ap}(\triangle_{m}(T))$$\sigma(\triangle_{m}(T))=\sigma_{ap}(\triangle_{m}(T))\cup[-\sigma_{ap}(\Delta_{m}(T))].$$k$$\triangle_{m}(T)$$\Delta_{m}(T)$$1\not\in\sigma_{p}(CTC)_{Z}$$\mathcal{H}.$$\cap$$T$$\cdots,$$n\leq j\leq m-1$

-complex symmetric operator and $F_{1}(T)\neq\{0\}$ , then $CF_{n}(T)$ is a nontrivial$m-1.$
invariant subspace for for $n=1$ , 2,

$m$

$T$

$\cdots,$

Example 5.3 Let $C$ be a conjugation operator given byon$\mathbb{C}^{3}$. If $R=I+N$ where$N=(\begin{array}{lll}0 1 00 0 30 0 0\end{array})$on$\mathbb{C}^{3}$$C(z_{1}, z_{2}, z_{3})=(\overline{z_{1}}, \overline{z_{2}}, \overline{z_{3}})$, then $N^{3}=0$ . Hence$\triangle_{4}(R)=\triangle_{4}(N)=6N^{*2}CN^{2}C=(\begin{array}{lll}0 0 00 0 00 0 54\end{array})\neq 0.$

is not a 4-complex symmetric operator. Hence $R=I+Ni_{\mathcal{S}}5$ -complex
Thussymmetric with $C.$
$R$

An operator
is said to be normaloid if $\Vert T\Vert=r(T)$ where $r(T)$ is$Cx\rangle=0$
is said to be isotropic if
the spectral radius of . A vector([14]). We next state some conditions for $(m+1)$ -complex symmetric operators to
be $m$-complex symmetric operators.
$T\in \mathcal{L}(\mathcal{H})$$T$$x\in \mathcal{H}$$\langle x,$

Theorem 5.4 Let
and be a conjugation on . Suppose $\triangle_{m+1}(T)=$is normaloid, and an eigenvector corresponding to every eigenvalue inis not isotropic. Assume that one of the following statements holds;
and athere exist(i) When $m$ is even, for every
sequenceandof unit vectors such that$T\in \mathcal{L}(\mathcal{H})$

$0,$

$C$

$\mathcal{H}$$\Delta_{m}(T)$$\sigma_{p}(\Delta_{m}(T))$$\mu\in\sigma_{ap}(\Delta_{m}(T))$$\{x_{n}\}$$\lambda\in\sigma(\Delta_{1}(T))$$|\lambda|^{m}=|\mu|$$\lim_{narrow\infty}\Vert(\triangle_{m}(T)-\mu)x_{n}\Vert=\lim_{narrow\infty}\Vert(\triangle_{1}(T)-\lambda)x_{n}\Vert=0.$31(ii) Whena sequence$mi\mathcal{S}$$\{x_{n}\}$odd, for everyof unit vectors such that$\mu\in\sigma_{ap}(\Delta_{m}(T))$

there exist $\lambda\in\sigma(T^{*}+CTC)$ andand$|\lambda|^{m}=|\mu|$$\lim_{narrow\infty}\Vert(\triangle_{m}(T)-\mu)x_{n}\Vert=\lim_{narrow\infty}\Vert((T^{*}+CTC)-\lambda)x_{n}\Vert=0.$

Then $\triangle_{m}(T)=0.$
Corollary 5.5 Let be a conjugation on and let
be a strict $(m+1)-$
complex ymmetric o$perator_{J}$ and an eigenvector corresponding to every eigenvalue
inbe not isotropic.
that one of the following statements holds;$m$(i) Whenis even, for every, there exist
and asequencesuch thatandof unit$C$$\mathcal{H}$$T\in \mathcal{L}(\mathcal{H})$$\mathcal{S}$$A_{\mathcal{S}sume}$$\sigma_{p}(\Delta_{m}(T))$$\mu\in\sigma_{ap}(\Delta_{m}(T))$$vector\mathcal{S}$$\{x_{n}\}$$\lambda\in\sigma(\triangle_{1}(T))$$|\lambda|^{m}=|\mu|$$\lim_{narrow\infty}\Vert(\Delta_{m}(T)-\mu)x_{n}\Vert=\lim_{narrow\infty}\Vert(\Delta_{1}(T)-\lambda)x_{n}\Vert=0.$(ii) When $m$ is odd, for every
a sequenceof unit vectors such that

there exist $\lambda\in\sigma(T^{*}+CTC)$ and$n=|\mu|$ and$\mu\in’\sigma_{ap}(\Delta_{\tau n}(T))_{f}$$\{x_{n}\}$$|\lambda|$}$\lim_{narrow\infty}\Vert(\Delta_{m}(T)-\mu)x_{n}\Vert=\lim_{narrow\infty}\Vert((T^{*}+CTC)-\lambda)x_{n}\Vert=0.$

Then$\Delta_{m}(T)$is not normaloid.

Theorem 5.6 For an operator
is $p-hyponormai_{f}$ then$T\in \mathcal{L}(\mathcal{H})$, let $\Delta_{2}(T)=0$ . If$T$is self-adjoint or$\triangle_{1}(T)=0.$$\triangle_{1}(T)$Corollary 5.7 Let $C$ be a conjugation operator on
and let $H$ and $K$ be selfadjoint operators. Suppose that $T=H+iK\in \mathcal{L}(\mathcal{H})$ satisfies $HCK=KCH$ and$CRC\geq R_{f}$ where $R=i(HK-KH)$ .
If $\Delta_{2}(T)=0$ , then $\Delta_{1}(T)=0.$$\mathcal{H}$

Theorem 5.8 If

is hyponormal and $\Delta_{m+1}(T)=0$ , then $ker(\Delta_{m}(T)-\lambda)\cap$$ker(\triangle_{1}(T)-\lambda)=\{O\}$for any nonzero$\triangle_{m}(T)$$\lambda\in \mathbb{C}.$

Corollary 5.9 Let be a conjugation on . Let $sef$-adjoint operators $H$ and $K$
satisfy $HCK=KCH$ and $CRC\geq R$ , where $R=i(HK-KH)$ . For an operator$T=H+iK$ , if $\Delta_{2}(T)=0$ , then $ker(\Delta_{1}(T)-\lambda)=\{O\}$ for any nonzero$C$$\mathcal{H}$$\lambda\in \mathbb{C}.$32

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Muneo Cho
Department of Mathematics
Kanagawa University
Hiratuka 259-1293
Japan-mail: chiyom01@kanagawa-u.ac.jp$e$Eungil Ko
Department of Mathematics
Ewha Womans University
Seoul 120-750
Korea-mail: eiko@ewha.ac.kr$e$

Ji Eun Lee
Department of Mathematics-Applied Statistics

Sejong University
Seou1143-747
Korea-mail: jieun7@ewhain.net; jieumlee7@sejong.ac.kr$e$