ON
2-HYPONORMAL
$\mathrm{O}\mathrm{P}\mathrm{E}\mathrm{R}\mathrm{A}\mathrm{T}O\mathrm{B}arrow\approx$成均館大学
(韓 )李
高等
(Woo
Young
Lee)
The purpose ofthis talk is to make a brief survey of recent research related to $2-\mathrm{h}_{-}\mathrm{v}.\mathrm{D}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{I}\mathscr{L}$
$\mathrm{O}_{\wedge}\mathrm{D}\mathrm{e}\mathrm{r}\mathrm{a}\dot{\mathrm{t}}\mathrm{o}\mathrm{r}\underline{\wedge-}$
.
Let
7#
and$\mathcal{K}$ be complexHilbert spaces, let$\mathcal{L}(\mathcal{H}, \mathcal{K})$ bethesetof bounded linearoperators from$\mathcal{H}$ to $\mathcal{K}$ and write
$\mathcal{L}(\mathcal{H}):=\mathcal{L}(\mathcal{H},\mathcal{H})$. An operator $T\in \mathcal{L}(\mathcal{H})$ is said to be normal if$T^{**}T=TT$,
hyponormal if$T^{*}T\geq TT^{*}$, and subnormal if$T=N|\mathcal{H}$, where $N$is normal on someHilbert space
$\mathcal{K}\supseteq \mathcal{H}$
.
Thus the operator$T$is subnormal if and only if there exist operators $A$ and $B$ such that$\hat{T}:=$ is normal, i.e.,
(0.1) $\{$
$[T^{*} T]:-\tau\tau*=A:=T^{*}TA*$
$A^{*}T=BA^{*}$
$[B^{*}, B]+A^{*}A=0$.
An alternativedescriptionofsubnormality is given by the Bram-Halmos criterion, which statesthat
anoperator$T$issubnormal if and only if
$\sum_{i_{\overline{\mathrm{J}}}},(\tau x_{j,t}\tau^{j}x)\geq 0$
for allfinite collections$x_{0},x_{1},$$\cdots$ ,$x_{k}\in \mathcal{H}$ ($[\mathrm{B}\mathrm{r}1,[\mathrm{c}_{\mathrm{o}\mathrm{n}}$, II.1.9]). It iseasytoseethat this is equivalent
to thefollowing positivity test:
(0.2)
$\geq 0$
(all $k\geq 1\rangle$.Condition (0.2) provides a measure of the gap between hyponormality and subnormality. In fact,
the positivity condition (0.2) for$k=1$ isequivalent tothe hyponormality of$T$, while subnormality
requires the validity of (0.2) for all $k$. Let $[A, B]:=$ AB–BA denote the commutator of two
operators $A$ and$B$, and define$T$to be $k$-hyponormal whenever the$k\cross k$ operatormatr\’ix
(0.3) $M_{k}(T):=([T^{*j},\dot{r}_{]}\backslash )_{i}^{\overline{\hslash}},j=1$
ispositive. Anapplication of the Choleski algorithm for operator matrices shows that the positivity
of(0.3)is equivalenttothe positivity of the$(k+1)\cross(k+1)$ operatormatrixin (0.2);theBram-Halmos
criterion canbethen rephrased assayingthat $T$is subnormal ifandonly if$T$is $k$-hyponormal for
every $k\geq 1$ ([CMXI). Now it is natural to ask whether$k$-hyponormaloperators admit an extension
1991 Mathemattcs Subject Classification. Primary$47\mathrm{B}20,47\mathrm{B}35,47\mathrm{B}37;\mathrm{s}_{\mathrm{e}}\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{a}\mathrm{l}\mathrm{y}47\mathrm{A}20$.
with one or more ofthe properties listed in (0.1). Recall $([\mathrm{A}\mathrm{t}],[\mathrm{c}\mathrm{M}\mathrm{X}],[\mathrm{c}\mathrm{o}\mathrm{S}])$ that $T\in \mathcal{L}(\mathcal{H})$ is
saidto beweakly$k$-hyponomd if
$LS((T, T^{2}, \cdots,T^{k})):=\{$$\sum_{j=1}^{k}\alpha_{j}\tau^{\mathrm{j}}$: $\alpha=(\alpha_{1}, \cdots, \alpha_{k})\in \mathbb{C}^{k}]\}$
consists entirely of hyponormal operators, orequivalently, $M_{k}(T)$ is weakly positive, i.e., ([CMXI)
(0.4)
$(M_{k(T)}, )\geq 0$
for$x\in \mathcal{H}$ and $\lambda_{0},$$\cdots,$$\lambda_{k}\in \mathbb{C}$
.
If$k=2$then$T$is said to be quadmtically hyponormal. Similarly,$T\in \mathcal{L}(\mathcal{H})$is said tobe polynomially
hyponormal if$p(T)$ ishyponormalforeveIypolynomial$p\in \mathbb{C}[z]$
.
It is known that $k- \mathrm{h}\mathrm{y}\mathrm{p}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}\Rightarrow$weakly$k$-hyponormal, but theconverseis not true in general. The classes of(weakly) k-hyponormal
operators have been studied inan attempt to bridge the gap between subnormality and
hyponor-mality$([\mathrm{C}\mathrm{u}\mathrm{l}\mathrm{l},[\mathrm{C}\mathrm{u}2],[\mathrm{c}\mathrm{F}\mathrm{l}], [\mathrm{C}\mathrm{F}2],[\mathrm{c}\mathrm{F}3],[\mathrm{c}\mathrm{L}\mathrm{L}\mathrm{l},[\mathrm{c}\mathrm{L}\mathrm{l}],[\mathrm{C}\mathrm{L}2],[\mathrm{C}\mathrm{L}3],[\mathrm{c}\mathrm{L}41,[\mathrm{c}\mathrm{M}\mathrm{x}], [\mathrm{D}\mathrm{P}\mathrm{Y}],[\mathrm{M}\mathrm{c}\mathrm{c}\mathrm{P}])$
.
Thestudy of this gap has been only $\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{n}_{\mathrm{y}}$ successful. For example, such a gap is not yet wel
de-scribed for Toeplitz operatorson the Hardyspace of the unit circle: in fact, evensubnormalityfor
Toeplitz operators has not yet been characterized (cf. [Hal], [Cowl], $[\mathrm{C}\mathrm{o}\mathrm{L}]$). For weighted shifts,
positive results appear in [Cul] and [CF3], althoughno concrete example ofaweightedshift which
ispolynomially hyponormal but not subnormal has yetbeen found (the existenceofsuchweighted
shiftswasestablished in [CP1] and $[\mathrm{C}\mathrm{P}2\mathrm{I})$
.
The following notionwasintroduced in [CL4].
1. Definition. An operator $T\in \mathcal{L}(\mathcal{H})$ is said to be weakly subnorrnal if there exist operators
$A\in \mathcal{L}(\mathcal{H}’,\mathcal{H})$ and $B\in \mathcal{L}(\mathcal{H}’)$ such that the first two conditions in (0.1) hold: $[T^{*},T]=AA^{*}$ and
$A^{*}T=BA^{*}$
.
The operator$\hat{T}$is said to
be..a
pariially normal estensionof$T$.
’.
Clearly,
(1.1) $\mathrm{s}\mathrm{u}\mathrm{b}\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}\Rightarrow$ weakly$\mathrm{s}\mathrm{u}\mathrm{b}\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}\Rightarrow$ hyponormal.
Howevertheconverses of both implications in (1.1) arenot trueingeneral (see $[\mathrm{C}\mathrm{L}4\overline{\mathrm{J}}$).
The following theorem providesa clue foramodel of2-hyponormal operators.
2.
Theorem
($[\mathrm{C}\mathrm{L}4$, Lemma 2.1]). $IfT\in \mathcal{L}(\mathcal{H})$ is2-hyponormal then$T$hasalinear (notnecessad$y$bounded) extension$\hat{T}$on$\mathcal{H}\oplus \mathcal{H}$ satisfying the equality$\hat{T}^{*}\hat{T}f=\hat{T}\hat{T}*f$
for
all$f\in \mathcal{H}$.
More precisely,(2.1) $\hat{T}:=(_{0}^{T}$ $[T^{*},T]^{\frac{1}{2}}\overline{S})$ : $\mathcal{H}\oplus Harrow \mathcal{H}\oplus \mathcal{H}.$
,
where$\overline{S}:ke\tau\{\tau*,$$\tau$]$\oplus nn[\tau^{*}, \tau]arrow \mathcal{H}$ is
defined
by(2.2) $\overline{S}f:=\{$
$[T^{*},T]^{\frac{1}{2}\tau}g$
if
$f=[T^{*},T]^{\frac{1}{2}}g$ with$g\in mn[\tau*,\tau]$Moreover
if
$\mathrm{I}^{\tau^{*},\tau}$] $ha\mathit{8}$dosed range ($e.g.$,
if
$[\tau*,\tau]$ isfinite
rank) then$Ti\mathit{8}$ weakly subnomal.Recall that $\underline{\mathrm{g}}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{n}$a bounded sequence of positive numbers $\alpha$
:
$\alpha_{0},$$\alpha_{1},$$\cdots$ (called $weight_{\mathit{8})}$, the(undateral) weighted
shift
$W_{\alpha}$ associated with $\alpha$ is the operatoron
$p_{(\mathbb{Z}_{+})}^{2}$ defined by $W_{\alpha}e_{n}:=$$\alpha_{n}e_{n+1}$ for all$n\geq 0$, where $\{e_{n}\}_{n=}^{\infty}0$ isthe canonicalorthonormalbasis for
$\ell^{2}$
.
It isstraightforward
to $\mathrm{c}\mathrm{h}\ovalbox{\tt\small REJECT}$ that $W_{\alpha}$ can neverbe nomal, and that $W_{\alpha}$ is hyponomd if and only if $\alpha_{n}\leq\alpha_{n+1}$ for
all$n\geq 0$
.
If$W_{\alpha}$ is a weighted shift with weight sequence $\alpha=\{\alpha_{n}\}_{n=0}^{\infty}$, then the $moment\mathit{8}$of $W_{\alpha}$are$\mathrm{u}\mathrm{s}\mathrm{u}\mathrm{a}\mathrm{u}_{\mathrm{y}}$ defined by$\beta_{0}:=1,$ $\beta_{n+1}:=\alpha_{n}\beta_{n}(n\geq 0)$ [Shi]; however, weprefer to reservethis term
for thesequence $\gamma_{n}:=\beta_{n}^{2}(n\geq 0)$
.
A criterion for$k$-hyponomalitycan be given in termsofthesemoments ([Cul, Theorem 4]): ifwe builda $(k+1)\cross(k+1)$ Hankel matrix$A(n;k)$ by
(2.3)
$A(n;k):=$
$(n\geq 0)$,then
(2.4) $W_{\alpha}$ is$k- \mathrm{h}\mathrm{y}\mathrm{P}^{\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{m}}\mathrm{a}1\Leftrightarrow A(n;k)\geq 0$ $(n\geq 0)$
.
In particular, for$\alpha$strictly increasing, $W_{\alpha}$ is 2-hyponormal if and only if
(2.5)
$\det\geq 0$
$(n\geq 0)$.
In [CL4] it was shown that 2-hyponormal weighted shifts are weakly subnormal operators
pos-sessingpartially normal extensions which
are
hyponormal.3.
$\mathrm{T}\mathrm{h}\infty \mathrm{r}\mathrm{e}\mathrm{m}$ ($[\mathrm{C}\mathrm{L}4$, Theorem 1.2]). Let $\alpha\equiv\{\alpha_{n}\}_{n=0}^{\infty}$ be a weight sequence.If
$W_{\alpha}i\mathit{8}$ a
2-hyponomdweighted
sh\’ifl
on $\mathcal{H}\equiv l^{2}(\mathbb{Z}_{+})$, then$W_{\alpha}\acute{l}\mathit{8}$ weakly $\mathit{8}ubnomal$.
Moreover, there exists apartially normal estension$\overline{W}_{\alpha}$
on$\mathcal{K}:=\mathcal{H}\oplus \mathcal{H}\mathit{8}uch$that
(i) $\overline{W}_{\alpha}$
is hyponomtal;
(ii) $\sigma(\overline{W}_{\alpha})=\sigma(W_{\alpha})$; and
(iii) $||\overline{W}_{\alpha}||=||W_{\alpha}||$
.
In particular,
if
$\alpha$ is $\mathit{8}t7\dot{\tau}ctlyinCrea\mathit{8}ing$then$\overline{W}_{\alpha}$ can be obtained$a\mathit{8}$
(3.1) $\overline{W}_{\alpha}=(_{0}^{W_{\alpha}}$ $[W_{\alpha W_{\beta}}^{*}, W_{\alpha}]^{\frac{1}{2}})$ on $\mathcal{K}:=\mathcal{H}\oplus \mathcal{H}$,
where $W_{\beta}$ is a weighted
shift
whose weightsequence $\{\beta_{n}\}_{n=}^{\infty}0$ given by$\beta_{n}=\alpha_{n}\sqrt{\frac{\alpha_{n+1}^{2}-\alpha_{n}^{2}}{\alpha_{n}^{2}-\alpha_{n-1}^{2}}}$ $(n=0,1, \cdots ; \alpha_{-1}:=0)$
.
In addition,
if
$W_{\alpha}$ is 3-hyponomal then$\overline{W}_{\alpha}$ canbe chosen to be weakly$\mathit{8}ubnomal$.
We now meet a natural question:Question A. $I\mathit{8}$ eve$7y2$-hyponomal operator weakly subnomal9
Towardsan affirmative answer wemust finda $\mathrm{p}-\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{n}_{\mathrm{V}}$
, normal extension
$\tilde{T}$
.
As acandidateone
mightsuggest, inviewof(2.1), that
$\hat{T}=$
.
where $S$ isa continuous linear extensionof$\overline{S}$
in Theorem 2. The keymissingstep isthen to show
that $S$ is bounded.
Ontheotherhand, do there exist hyponormal weightedshifts whichare notweakly
subnormal?
Toanswerthis question, we first giveanouterpropagation property of weakly subnormalweighted
shifts.
4. Theorem ($[\mathrm{C}\mathrm{L}4$, Theorem 4.3]). Let $T\equiv W_{\alpha}$ be a weighted $\mathit{8}hifl$ with weight sequence $\alpha=$
$\{\alpha_{n}\}_{n=0}^{\infty}$
.
Assume that$T$ is weakly subnomal.If
$\alpha_{n}=\alpha_{n+1}$
for
some $n\geq 0$ then$\alpha_{n+k}=\alpha_{n}$for
all$k\geq 1$
.
5. Example ($[\mathrm{C}\mathrm{L}4$, Example 4.4]). With the aidofTheorem 4
we
can findexamplesofoperatorswhich
are
hyponormal (even quadraticallyhyponormal) but not weakly subnormal: for example, if$\alpha_{0}=\alpha_{1}=\sqrt{\frac{2}{3}}$, (cf. [Cul, Proposition 7]),
then $W_{\alpha}$ isquadratically hyponormal but not weakly subnormal. $\square$
Nowone might expect an innerpropagation (and hence$\mathrm{f}\mathrm{u}\mathrm{U}$propagation) forweakly subnormal
weighted shifts. But we don’t know if this is the case. In fact we were unable to decide ifevery
weakly subnormal weighted shift is 2-hyponormal. Ifa weighted shift $T$ has a partially normal
extension$\hat{T}$
oftheform (3.1) thenwe can see ($[\mathrm{C}\mathrm{L}4$, the proofofTheorem 1.2]) that $T$ should be
2-hyponormal. But it is not evident that whenever a hyponormal weighted shift $T$has a partially
normalextension $\hat{T},\acute{\dot{T}}$ should be of the form (3.1) up to unitaIy equivalence. Thuswe would like
to posethe following:
Question B. $I\mathit{8}$ everyweakly $\mathit{8}ubnomal$ weighted
shift
2-hyponomal9Let’s turnourattention to 2-hyponormality ofToeplitz operators. Recallthat the Hilbert space
$L^{2}(\mathrm{T})$ has a canonical orthonormal basis given by the trigonometric functions $e_{n}(z)=z^{n}$, for all
$n\in \mathbb{Z}$, and that the Hardy space $H^{2}(\mathrm{T})$ is the closed linear spanof$\{e_{n} : n=0,1, \ldots\}$
.
An element$f\in L^{z}(\mathbb{T})$ issaid to be analytic if$f\in H^{2}(\mathrm{T})$, and $\mathrm{c}$ -analytic if$f\in L^{2}(\mathrm{T})\ominus H^{2}(\mathrm{T})$. If$P$denotes
theorthogonal projection from$L^{2}(\mathrm{T})$ to $H^{2}(\mathrm{T})$ and $J$is the unitaryoperatorfrom$L^{2}(\mathrm{T})\ominus H^{2}(\mathrm{T})$
onto $H^{2}(\mathrm{T})$, then for every$\varphi\in L^{\infty}(\mathbb{T})$, the operator$T_{\varphi}$ on$H^{2}(\mathrm{T})$definedby
$T_{\varphi}g:=P(\varphi g)$ ($g\in H^{2}$(T.).)
is called the Toeplitzoperatorwith symbol$\varphi$
.
It is well known that analytic Toeplitz operators aresubnormal.
The study of subnormal Toeplitz operators was originated from $\mathrm{P}.\mathrm{R}$
.
Halmos’s Problem 5 (cf.question was answered in the negative byC. Cowen and J. Long [$\mathrm{c}_{0}\mathrm{L}|$
.
But it is still open whichToeplitz$operator\mathit{8}$ are$\mathit{8}ubnomalg$ The hyponormalityofToeplitz operators has been studied by M.
Abrahamse [Ab], C. Cowen [$\mathrm{c}_{\mathrm{o}\mathrm{w}}11\cdot[\mathrm{c}_{\mathrm{o}\mathrm{w}2]}$, P. Fan [Fa], C. Gu [Gu], T. Ito and T. Wong $[\mathrm{I}\mathrm{t}\mathrm{W}]$, T. Nakazi and K. Takahashi [NT], D. Yu [Yu], K. Zhu [Zh], D. Farenick, the author, and his
collabora-tors (cf. $[\mathrm{F}\mathrm{L}1],[\mathrm{F}\mathrm{L}2],[\mathrm{c}\mathrm{L}1],[\mathrm{H}\mathrm{K}\mathrm{L}],[\mathrm{K}\mathrm{L}]$). An elegant theorem ofC. Cowen [Cow3] characterizesthe
hyponormality ofaToeplitz operator$T_{\varphi}$ on$H^{2}(\mathrm{T})$ by properties of the symbol$\varphi\in L^{\infty}(\mathrm{T})$. K. Zhu
[Zh] reformulated Cowen’s criterion and then showed that the hyponormality of$T_{\varphi}$ with polynomial
symbok$\varphi$canbe decided by a method based on the classical interpolation theorem of I. Schur [Sch].
Now itseemsto beinterestingto understand thegapbetween$k$-hyponormality and subnormality
for Toeplitz operators. As a candidate for the first question in this line we posed the $\mathrm{f}\mathrm{o}\mathrm{l}\mathrm{l}0\check{\mathrm{w}}$
ing
$\langle$$[\mathrm{C}\mathrm{L}1],[\mathrm{C}\mathrm{L}3_{\lrcorner}^{]\backslash }j$:
Question C. Is every 2-hyponomd Toeplitz operator$\mathit{8}ubnomal^{i)}$
In [CL3], thefollowing wasshown:
6. Theorem ([CL3, Corollary 6]).
If
$T_{\varphi}$ is 2-hyponomal andif
$\varphi$ or$\overline{\varphi}$ isof
boundedtype $(i.e.,$ $\varphi$or$\overline{\varphi}i\mathit{8}$a quotient
of
two analyticfunctions) then$T_{\varphi}$ is nomal oranalytic.Inviewof Theorem 6, it would be interesting to consider which 2-hyponormal$\mathrm{T}_{\mathrm{o}\mathrm{e}}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}_{\mathrm{Z}}$operators
aresubnormal. Thefirstinquiry involves the self-commutator. Subnormaloperatorswith finite rank
self-commutators have been studied by manyauthors ([A1], $[\mathrm{M}_{\mathrm{C}}\mathrm{C}\mathrm{Y}\mathrm{a}]$
,
[Mo], [OTT], [Xil], [Xi2]). In1975, I. Amemiya, T. Ito and T. Wong [AIW] showed that if$T_{\varphi}$ is a subnormal Toeplitz operator
with rank-one self-commutator then$\varphi$ is alinear function ofa Blaschke productof degree 1. More
generally, B. Morrel [Mo] showed that a pure subnormal operator with rank-one self-commutator is
unitarily equivalent toa linearfunction of the unilateral shift. Very recently, in [CL4], itwasshown
that everypure 2-hyponormal operator with rank-one self-commutator is a linear function of the
unilateralshift. Ontheotherhand,J. $\mathrm{M}\mathrm{c}\mathrm{C}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{h}\mathrm{y}$ andL.Yang [$\mathrm{M}_{\mathrm{C}}\mathrm{C}\mathrm{Y}\mathrm{a}|$haveclassifiedall rationally
cyclicsubnormal operators with finite rank self-commutat$o\mathrm{r}\mathrm{s}$. However it is still open whicharethe
puresubnormal operatorswith finite rankself-commutator. Related to this, in [CL3] weformulated
thefollowing:
Question D.
If
$T_{\varphi}\dot{u}$ a 2-hyponomal Toeplitz operator withnonzerofinite
rank self-commutator,does it
follow
that$T_{\varphi}i\mathit{8}$ analytic 9Onthe other hand, there exists agapbetween 2-hyponormality and quadratic hyponormalitv for
weighted shifts (cf. [Cul]). However we have not been able to decide whether there exists a gap
between2-hyponormality and quadratic hyponormality for Toeplitz opemtors.
Question E. $Doe\mathit{8}$there existaquadmtically hyponomal Toeplitz operator which$i\mathit{8}$not2-hyponomal9
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DEPARTMENTOF MATHEMATICS,$\mathrm{S}\mathrm{u}\mathrm{N}\mathrm{c}\mathrm{K}\mathrm{Y}\mathrm{u}\mathrm{N}\mathrm{K}\mathrm{w}\mathrm{A}\mathrm{N}$ UNIVERSITY, SUWON440-746, KOREA
