YOKOHAMA MATHEMATICALJOURNAL VOL. 26, 1978
ESSENTIALLY NORMALOID AND ESSENTIALLY
CONVEXOID OPERATORS*
S. M. PATEL
(Received May 26, 1977)
ABSTRACT.In this note some elementry facts on essentially normaloid and
essentially convexoid operators are established.It is shown that (i) a nomaloid
operator need not be essentially normaloid and vice versa,(ii) a paranormal
operator is not essentially convexoid, (iii) the class of essentially normaloidoperators is a closed nowhere dense subset of the algebra of operators, and
(iv) for an essentially convexoid (essentially normaloid, essentially spectraloid)operator$T$, there exists a compact operator$K$ such that $T+K$ is convexoid
(normaloid, spectraloid).
In what follows, an operator on a complex Hilbert space $H$ will be meant a
bounded linear transformation on$H$.Let $B(H),$ $K(H)$ denote the Btach algebra
of operators on$H$, the two sided ideal of compact operators on$H$ and the Calkin
algebra of$H$.The notation$\pi(T)$ is used for the canonical image of$T$ in the
Calkin algebra.The essential spectrum, the essential numerical range, the essential
spectral radius and the essential unmerical radius of$T$ will be denoted by$\sigma.(T)$,
$W.(T),$ $r(T)$ and$|W_{e}(T)|$ respectively.
An operator$T$ is called essentially normaloid if $ r.(T)=\Vert\pi(T)\Vert$ ; essentially$\infty n-$
vexoid if$\infty n\sigma_{e}(T)=W.(T)$ and essentially spectraloid if $r.(T)=|W.(T)|$.
The auth6r [4] has shown that if$T$ is reduction-spectraloid (reduction-normaloid)
then$T$ is essentially spectraloid (essentially normaloid).In particular, every re-
duction-convexoid operator is essentially $\omega nvexoid$. However the convexoidity alone
is not suﬃcient to guarantee the essential$\infty nvexoidity$ (see Luecke [3, $Threm$
$7])$.Similarly, as we shall show in Theorem 2, the normaloidity does not always
imply the essential normaloidity.First we establish the following $threm$ which
is analogous to Theorem 5 of [3].
Theorem 1. If
$T=\left(\begin{array}{ll}A & C\\‘ & \\0 & B\end{array}\right)$
$*$$rs$ (1970) subiect classiﬁcation.Primary 4B20, $47B05$.
$\cdot$
56S. M. PATEL
on$H\oplus H$, where$C$ is an (essentially) invertible oPerator, then$T$ is not (essentially)
normaloid.
Proof.(I).Since$C$ is invertible, there exists $M>0$ such that$\Vert Cx\Vert\geq M\Vert x\Vert$
and$||C^{*}x\Vert\geq M|x\Vert$ for all$x$ in$H$.For a unit vector$x$ in$H,$$T\left(\begin{array}{l}0\\x\end{array}\right)=\left(\begin{array}{l}Cx\\Bx\end{array}\right)$ and
hence$\Vert T\Vert\geq||Cx\Vert+\Vert Bx\Vert\geq M^{2}+\Vert Bx\Vert$ .From this we get
(1)$\Vert T||>\Vert B||$ .
Similarly, as
$T^{*}\left(\begin{array}{l}x\\0\end{array}\right)=\left(\begin{array}{l}A^{*}x\\C^{*}x\end{array}\right)$ , we obtain
\langle 2)$\Vert T\Vert>\Vert A\Vert$ .
Since$\sigma(T)\subseteq\sigma(A)\cup\sigma(B)$, we have
\langle 3)$r(T)\leq\max\{r(A), r(B)\}$ .
From (1), (2) and (3), we infer that$ r(T)\leq\max\{r(A), r(B)\}\leq\max\{||A||, \Vert B\Vert\}<\Vert T\Vert$ ,
showing that$T$ is not normaloid.
(II).The proof is analogous to given in Part II of Theorem 5 [3].
Theorem 2.There exists a normaloid operator which is not essentially nor-
$Jn\dot{a}loid$ and vice versa.
Proof.Let$H=H_{1}\oplus H_{2}\oplus H_{3}$, where each$H_{i}$ is inﬁnite dimensional.Let
$A=\left(\begin{array}{ll}0 & I\\0 & 0\end{array}\right)$
be an operator on$H_{1}\oplus H_{2}$.Let $N=diag\{1,1/2,1/3\cdots\}$ be an operator on$H,$.If
$T=A\oplus N$, then $r(T)=r(N)=1$ and$||T\Vert=\max\{\Vert A\Vert, \Vert N\Vert\}=1$; thus$T$ is normaloid.
Since by $Thoem1$, A is not essentially normaloid,$T$ is not essentially normaloid.
To show the existenoe of an essentially normaloid but non-normaloid operator,take any non-normaloid compact operator.
Ando [1] has$\infty nstructed$ a paranormal operator which is not $nvexoid$.This
raises the following question:Does there exist a paranormal operator which is not
essentially $\infty nvexoid$?Our next result answers this question in negative.As an
immediate consequence of this, we get that the class of essentially hyponormalooerators is a prooer subset of the class of essentially paranormal operators(and
hence of essentially normaloid operators).
Theorem 3.There exists a paranormal oPerator which is not essentiallv
convexoid.
ESSENTIALLY NORMALOID AND ESSENTIALLY CONVEXOID OPERATORS57
Proof.Let$U$ be the unilateral shift of inﬁnite multiplicity.Let $A=U+1$.
Then by [1], the operator
$T=\left(\begin{array}{ll}A & (A^{*}A-AA^{*})^{1/2}\\0 & 0\end{array}\right)$
deﬁned on$H\oplus H$ is paranormal.We claim that$T$ is not essentially convexoid.
First observe that Bdry$\sigma.(A)\subseteq\sigma_{e}(T)\subseteq\sigma(A)$ and so con $\sigma_{e}(A)=n\sigma.(T)$.
Suppose to the contrary that$T$ is essentially convexoid.Then$||(\pi(T)-\lambda)^{-1}||\leq$
$1/d$(
$\lambda$, con $\sigma.(T)$) $=1/d$(
$\lambda$, con
$\sigma_{*}(A)$) for all$\lambda\not\in con\sigma_{e}(A)=con\sigma_{*}(T)$.In particular,
$||(\pi(T)+1)^{-1}||\leq 1/d$(
$1$, con$\sigma_{e}(T)$) $=1$.Let
$P=\left(\begin{array}{ll}1 & 0\\0 & 0\end{array}\right)$ be the orthogonal projection on
$H\oplus O$.Since $B(H)/K(H)$ is a
$C^{*}$-algebra, there exists a Hilbert space$H_{0}$ such that
$B(H)/K(H)$ is isometrically isomorphic to a closed, selfadjoint subalgebra of$B(H_{0})$.
Let$v$ be this isometric isomorphism.Then$v\cdot\pi(P)$ is an orthogonal projection in
$B(H_{0})$.Let$H_{0}=M\oplus M^{\perp}$, where$M$ is the range of$v\cdot\pi(P)$.Relative to this
decomposition of$H_{0}$,
$v\cdot\pi(T)=\left(\begin{array}{ll}A_{1} & C_{1}\\0 & 0\end{array}\right)$
$(v\cdot\pi(T)+1)^{-1}=\left(\begin{array}{ll}(A_{1}+1)^{-1} & -(A_{1}+1)^{-1}C_{1}\\0 & 1\end{array}\right)$ .
Therefore
$\Vert(\pi(T)+1)^{-1}\Vert=\Vert(v\cdot\pi(T)+1)^{-1}\Vert>1$ ,
a contradiction.Thus$T$ is not essentially convexoid.
Theorem 4.The class of essentially normaloid operators is a closed nowhere
dense subset of $B(H)$.
Proof.First we note that$\Vert\pi(T)\Vert=\Vert v\cdot\pi(T)\Vert$ and$\sigma_{e}(T)=\sigma(v\cdot\pi(T))$, where$v$
is the isometric (algebra) embedding of the Calkin algebra in$B(H_{0}),$$H_{0}$ being a
Hilbert space.Therefore$T$ is essentially normaloid if and only if$v\cdot\pi(T)$ is nor-
maloid.From this we conclude that the class of essentially normaloid operators
is closed.To complete the proof, it now suﬃces to show that for an essentially
normaloid operator$T$, there exists a sequence$\{T_{n}\}$ of non-essentially normaloid
operators converging to$T$.
Since$\pi(T)$ is normaloid, there exists a $mplex$ number
$\lambda$ in$\sigma.(T)$ such that
$|\lambda|=\Vert\pi(T)\Vert$ .Let$\{x_{n}\}$ be a sequence of orthonormal vectors in$H$ for which$\Vert(T-$
$\lambda)x,\Vert+\Vert(T-\lambda)^{*}x_{n}\Vert\rightarrow 0$.By Stampﬂi’s corollary to $Threm2[5],$$T$ is unitarily
equivalent, via unitary operator$U$, to$T\oplus\lambda+K$ on$H\oplus H_{1}$, where
$H_{1}$ is a seoerable
58S. M. PATEL
Hilbert space and $K$ is a compact operator on$H\oplus H_{1}$.Let
$B=\left(\begin{array}{ll}0 & 1\\0 & 0\end{array}\right)$ be an operator
on$H_{1}$ and let$S_{\#}=T\oplus(\lambda-B/n)+K$ be an operator on$H\oplus H_{1}$.We claim that$S$,
is not essentially normaloid.To this end, observe that$\Vert\pi(\lambda-B/n)\Vert>|\lambda|=\Vert\pi(T)||=$
$r_{e}(\lambda-B/n)$.Therefore
(1)$\Vert\pi(S_{n})||=maT\{\Vert\pi(T)\Vert, \Vert\pi(\lambda-B/n)\Vert\}=\Vert\pi(\lambda-B/n)\Vert$
$r.(S_{l})=\max\{r.(T), r.(\lambda I-B/n)\}$
$=\max\{\Vert\pi(T)\Vert, r.(\lambda-B/n)\}$
$=r_{e}(\lambda-B/n)$ .
(2)$ r.(S_{*})<\Vert\pi(\lambda-B/n)\Vert$ .
From(1) and (2), we conclude that$S$, is not essentially normaloid.Also
$\Vert T-US_{*}U^{*}\Vert=\Vert S-S_{\sim}\Vert=1/n\Vert B\Vert=1/n\rightarrow 0$ as
$ n\rightarrow\infty$ .Thus$\{T,\}=[US,U^{*}$} is the
desired sequence of operators.
Our ﬁnal result establishes a conjecture due to Lueck [3, Conjecture 2].
Theorem 5.(i) If
$T$ is an essentially convexoid operator, then there exists a
comPact operator $K$ such that $T+K$ is convexoid.
(ii) If
$T$ is essentially normaloid (essentially spectraloid), then there exists a
comPact operator $K$ such that $T+K$ is normaloid (spectraloid).
Proof.(i)To prove the result, we note that for any operator$T$, there exists
a compact operator$K$ such that
(1)$ 11T+K+\lambda\Vert=\Vert\pi(T+\lambda)\Vert$
for all complex numbers [2].Moreover
(2)$W.(T)=\bigcap_{\lambda eC}\{\mu:|\mu-\lambda|\leq||\pi(T-\lambda)\Vert\}$
Combining (1) and (2), we get
$W.(T)=\bigcap_{\lambda eG}\{\mu:|\mu-\lambda|\leq\Vert T+K-\lambda||\}$
$=\overline{W(T+K)}$ .
Therefore if$T$ is essentially convexoid, then con $\sigma(T+K)\supseteq con\sigma.(T)=W.(T)=$
$\overline{W(T+K)}$, which shows that $T+K$ is convexoid.Parts (i) and (ii) are now obvious.
References
[ 1]T. Ando:Paranormality does not imply convexoicl properry.(Private communication).
ESSENTIALLY NORMALOID AND ESSENTIALLY CONVEXOID OPERATORS59
[2]C. K. Chui, P. W. Smith, J. D. Ward and D. A. Legg:On a questim of OZsen con-
cerning eompact perturbations of operators.Notices Amer. Math. Soc., 23(1) (1976),
A.164.
[3]G. R. Luecke.Essentially$(G_{1})$ operators and essentially convexaid operators on Hilbert
space.Illinois J. Math., 19(3) (1973), 389-399.
[4]S. M. Patel:A note on spectraloid operators.Glasnik Matematicki,11(31) (1976),
319-324.
[5]J. G. Stampﬂi:Compact$ perturktion\epsilon$, normal eigenvalues, and a prouem of Salinas.
J. London Math. Soc., (2) 9 (1974), 165-175.
Faculty of MathematicsUniversity of DelhiDelhi-110007, India
Department of MathematicsSardar Patel UniversityVallabh Vidyanagar388120, Gujarat, India