YOKOHAMA MATHEMATICALJOuRNAL VOL. 26, 1978
NORMAL APPROXIMATE SPECTRUM OF OPERATORS
S. N. RAI
(Received October 3, 1977)
ABSTRACT.Putnam [5] considered
$\lambda$ and$T$ as accessible point of spectrum
of$T$ and semi-normal operator($(TT^{*}-T^{*}T)=D\geq 0$ or$D\leq 0$)$resoectively$
and proved some inequalities, which I have proved again in more generalform for
$\lambda$ as a normal approximate spectrum and$T$ as any bounded linear
operator on a Hilbert space.
1.Only bounded linear operators on a Hilbert space are to be considered.
A bounded linear operator$T$ is called semi-normal if $TT^{*}-T^{*}T=D,$$D\geq 0$ or
$D\leq 0$.Clearly any normal operator is also semi-normal.It is easy to see that
the converse is also true in case the space is ﬁnite dimensional.For if, say,$ D\geq$
$0$, its eigenvalues are non-negative while their sum is the trace of$D$, which is
zero.Hence all eigenvalues are
$0$ and so $D=0$:In the inﬁnite dimensional case
however it is possible that an operator be semi-normal without being normal.In
fact, any isometric but not unitary operator$V$ has this property; for$ VV^{*}-V^{*}V\leq$
$0,$$\neq 0$.On
$l^{2}$ the operator$T$ given by the matrix$T=(a_{i.\dot{g}})$ with$a_{i+1.i}=1$ and
$a:.j=0$ otherwise$(i, j=1,2\cdots)$ is such an operator.
The above statement shows that an operator be semi-normal without being
normal, it can also be proved in the case of$T$ being unbounded.First we deﬁned
the notions of hyponormality and semi-normality for not necessarily boundedoperators.
An operator$T$ is called hyponormal if it is closed, densely deﬁned and satisﬁes
the conditions:
$ D_{T}=D_{T}\cdot$ ,
$\Vert T^{*}x\Vert\leq\Vert Tx\Vert$ for$x\in D_{T}$, where$D_{T}$ is domain of$T$.
An operator$T$ is called semi-normal if it is closed, densely deﬁned and if$T$
$T^{*}$ is hyponormal.It is clear that every normal operator is hyponormal and
therefore semi-normal.Moreover the above deﬁnitions are extensions of the deﬁ
$\cdot$
nitions of hyponormality and semi-normality for bounded operators.
Let$H=l^{2}$ and let$T$ be the inﬁnite matrix
$(a_{i\dot{g}})_{\ell,j=1}^{\infty}$ ,
78S. N. RAI
where$a_{i,i+1}=i$ for $i=1,2,$$\cdots$ , and$a_{ij}=0$ for $j\neq i+1$.We will show that
$T^{*}$ is
hyponormal, it is clear that$T$ is closed and densely deﬁned.It remains therefore
to show that$D_{T}=D_{T^{\prime}},$$\Vert Tx\Vert\leq\Vert T^{*}x\Vert$ for each$x$ in$D_{T}$ and that there exists vector
in$D_{T}$ for which$\Vert Tx\Vert\leq\Vert T^{*}x\Vert$ .
$D_{T}=\{x=(\xi_{1}, \xi_{2}, \cdots);\sum_{i=1}^{\infty}(i-1)^{2}|\xi_{i}|^{2}<\infty\}$ .
$D_{T^{*=}}\{x=(\xi_{1}, \xi_{2}, \cdots);\sum_{l=1}^{\infty}i^{2}|\xi_{i}|^{2}<\infty\}$ .
For each$x\in D_{T;}x\neq 0$ we have
$\Vert Tx\Vert^{2}=\sum_{i=2}^{\infty}(i-1)^{2}|\xi_{i}|^{2}<\sum_{i=1}^{\infty}i|\xi_{i}|^{2}$ .
Moreover, since$i^{2}<2(i-1)^{2}$ for$i\geq 2$,theinequality$\sum_{i=2}^{\infty}(i-1)^{2}|\xi_{i}|^{2}<\infty$implies
$\sum_{\ell=1}^{\infty}i^{g}|\xi_{i}|<\infty$.The semi-normality of$T$ is proved.
$\theta$ is real it is clear that$e^{it}T$ is also semi-normal whenever$T$ is, if $T=H+iJ$
is replaced by$e^{i\theta}T=H_{\theta}+iJ_{\theta}$.Then
$H_{\theta}=\frac{1}{2}(e^{i\theta}T+e^{-it}T^{*})$
$J_{\theta}=(\frac{1}{2\dot{\iota}})(e^{it}T-e^{-i\partial}T^{*})$ .
$T_{\theta}=e^{\ell\partial}T$for real
$\theta$ .
$H_{t}=\frac{1}{2}(T_{\theta}+T_{\theta}^{*})$ .
It is seen that
$H_{\theta}$ is the real or imaginary part of$T$ according as$\theta=0$ or
$\theta=-\pi/2$.
If $\lambda\in sp(T)$ then
$\lambda$ is called accessible if there exists a sequence
$\{\lambda_{n}\},$$\lambda_{n}\not\in sp(T)$,
satisfying
$\lambda_{n}\rightarrow\lambda$ as
$ n\rightarrow\infty$ .A complex number
$\lambda$ is an approximate proper value
of$T$ provided that
$\lambda$ and$T$ satisfy
(1.1)$\Vert Tx_{n}-\lambda x_{n}\Vert\rightarrow 0$$(n\rightarrow\infty)$
for a sequence$\{x_{n}\}$ of unit vectors.Furthermore, if
$\lambda$ and$T$ satisfy (1.1) and
(1.2)$\Vert T^{*}x_{*}-\overline{\lambda}x_{n}\Vert\rightarrow 0$$(n\rightarrow\infty)$ ,
$\lambda$ is called a normal approximate proper value of$T$.The normal approximate
spectrum denoted by$\Pi_{n}(T)$ is deﬁned to be the set of all normal approximate
proper values of$T$.
NORMAL APPROXIMATE SPECTRUM OF OPERATORS79
For a (continuous linear) operator$T$ on a Hilbert space, we use the following
notation and terminology: spectrum $sp(T),$$\ovalbox{\tt\small REJECT} ntinuous$ spectrum $C(T)$, approximate
point spectrum $\Pi(T)$, point spectrum $p(T)$, spectral radius$r(T)=\sup\{|\lambda|:\lambda\in sP(T)\}$,
boundary
$\partial$, numerical range$W(T)$ is
$\ovalbox{\tt\small REJECT} nvex$, and$\Sigma T\subset ClW(T)(\Sigma=convex$ hull
of the spectrum, $Cl=closure$), if $\Sigma T=ClW(T)$ then$T$ is called convexoid operator
and we say that an operator$T$ is restriction convexoid if the restriction of$T$ to
every invariant subspace has property convexoid.
2.Putnam [5] proved the following theorem:
Theorem A.Let$T$ be bounded and satisfy$TT^{*}-T^{*}T=D\geq 0$ and let
$\lambda=re^{-i2}$
$(r\geq 0)$ be an accessible point of $sp(T)$.Then
$(\max H_{\theta})^{2}\geq\min TT^{*}$
$|r-\max H_{\theta}|\leq[(\max H_{\theta})^{2}-\min TT^{*}]^{1/}$ .
We prove the Putnam inequalities for bounded linear operator taking
$\lambda$ in the
normal approximate spectrum.Actually, we will prove the following.
Theorem 2.1. If
$T$ is any operator and
$\lambda\in\Pi_{n}(T)$ ,
$\lambda=re^{-i\theta}$$(r\geq 0)$ ,
(2.1)$(\max H_{\theta})\geq r\geq(\min T^{*}T)^{1/2}$
(2.2)$|r-\max H_{\theta}|\leq[(\max H_{\theta})^{2}-\min T^{*}T]$ .
Proof.Since$\lambda\in\Pi_{n}(T)$ , there exists a sequence
$\{x_{\iota}\}$ of unit vectors such that
$\Vert(T-\lambda I)x_{n}\Vert\rightarrow 0$and
$\Vert(T^{*}-\overline{\lambda}I)x_{n}\Vert\rightarrow 0$ .
We get
$(T-\lambda I)x_{n}\rightarrow 0$and
$(T^{*}-\overline{\lambda}I)x_{n}\rightarrow 0$ .
Therefore
$((T-\lambda I)x_{n}, x_{n})\rightarrow 0$ .
Similarly
$((T^{*}-\overline{\lambda}I)x_{n}, x_{n})\rightarrow 0$ .
It follows that
$(e^{i\theta}Tx_{n}, x,)+(e^{-i\theta}T^{*}x_{n}, x_{*})\rightarrow\lambda e^{i\theta}+\overline{\lambda}e^{-i\theta}$
$\frac{1}{2}((e^{i\theta}T+e^{-i\theta}T^{*})x_{n}, x_{\iota})\rightarrow\frac{1}{2}(r+r)$
80S. N. RAI
$(H_{\theta}x_{n}, x_{n})\rightarrow r$ ,
and therefore
$H_{\theta}\geq r$ .
$0=\lim((T-\lambda I)x_{*}, Tx_{n})$
$0=\lim(T^{*}Tx_{n}, x,)-\lambda(x_{*}, Tx_{n})$ ,
we get
$\lim(T^{*}Tx_{n}, x_{n})=\lambda\overline{\lambda}=r^{2}$
$T^{*}T\leq r^{2}$
max $H_{\theta}\geq r\geq(\min T^{*}T)^{1/2}$ .
$(T-\lambda I)^{*}(T-\lambda I)=(T^{*}-\overline{\lambda}I)(T-\lambda I)$
$=(T^{*}T-\overline{\lambda}T-\lambda T^{*}+\lambda\overline{\lambda})$
$=T^{*}T-2rH_{\theta}+r^{1}$ ,
one has
$\Vert(T-\lambda I)x_{n}\Vert=(T^{*}Tx_{n}, x_{*})-2r(H_{\theta}x_{n}, x_{n})+r^{2}$ .
min $T^{*}T\leq(T^{*}Tx_{n}, x_{n})=\Vert(T-\lambda I)x_{n}\Vert+2r(H_{\theta}x_{n}, x_{n})-r^{g}$ .
Letting$ n\rightarrow\infty$
min $T^{*}T\leq 2r$ max
$H_{\theta}-r^{2}$
$\leq(\max H_{\theta})^{2}-(\max H_{\theta}-r)^{2}$
$|r$-max $H_{\theta}|\leq[(\max H_{\theta})^{2}$-min $T^{*}T]^{1/2}$
Theorem 2.2.Let$T$ be a restriction convexoid operator and
$\lambda=re^{-l\theta},$$(r\geq 0)\in$
$sp(T)$ is ﬁnite, then (2.1) and (2.2) hold.
Before giving the proof of theorem 2.2 we need the following lemmas:
Lemma A [3, Theorem 1]: If
$\lambda$ belongs to$\partial W(T)$ and$\Pi(T)$ then$\lambda\in\Pi,,(T)$,
where$\partial W(T)$ is the frontier of numerical range of
Lemma$B$ [$4$, Theorem 3]: If
$\lambda$ is a normal apprOximate proper values of
$A$, then there exists a character
$\phi$ on the
$C^{*}$-algebra$U$ generated by$A$ and
which satisﬁes
$\phi(A)=\lambda$ .
Lemma$C$ [
$1$, Lemma 2]: If
$T$ is restriction-convexoid and
$\lambda$ is an isolated
point of $sp(T)$, then
$\lambda$ is an eigenvalue.
NORMAL APPROXIMATE SPECTRUM OF OPERATORS81
Proof of Theorem 2.2.By lemma $Csp(T)=p(T)\subset W(T)$ since$T$ is convexoid,
$ClW(T)=\Sigma(T),$$\Sigma(T)\subset W(T)$, thus$W(T)$ is closed,$W(T)=\Sigma(T)$ and the extreme
points of$W(T)$ belong to $sp(T)$.Therefore, if
$\lambda$ be the extreme wint of$W(T)$
then $\lambda\in sp(T)\cap\partial W(T)\subset\Pi,(T)$ by Hildebrandt’s theorem [3] Satz 2.Hence we have
$\lambda e\Pi.(T)$ .
Now, from
$Th\ovalbox{\tt\small REJECT} rem2.1$ we get the result.
Remark.If$T$ is hyponormal and$\lambda=re^{-i\theta}(r\geq 0)$ and $sP(T)=C(T)$, then (2.1)
and (2.2) hold for$T$.
Proof.Since $sp(T)=C(T)$ implies that there exists a sequence of unit vectors,
say$\{x,\}$, such that
$\Vert Tx_{n}-\lambda x_{\hslash}\Vert\rightarrow 0$$(n\rightarrow\infty)$ .
From hyponormality, we get
$\Vert T^{*}x_{f*}-\overline{\lambda}x,\Vert\rightarrow 0$$(n\rightarrow\infty)$ .
Now from Theorem 2.1 we get the result.
References
[1]S. K. Berberian:Some conditions on an operator implying normality.Math. Ann.,
184 (1970), 188-192.
[2]M. Fuji and R. Nakamoto:On normal approximate Spectrum II.Proc. Japan Acad.,
48 (1972), 297-301.
[3]S. Hilderbrandt:Vber den$numeri\epsilon chen$ Werteberewh
$ eine\epsilon$ operators.Math. Ann.,
163 (1966), 230-247.
[4]Kasahara and Takai:Approximate proper value and Character of
$C^{*}\cdot algebra$.Proc.
Japan Acad., 48 (1972), 91-93.
[5]C. R. Putnam:On semi-normal operators.Paciﬁc J. Math., 7 (1957), 1649-1652.
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