Abstract
ABSTRACT. For <p in L°°(dD), let <p = f + g where / and g are in H2.
In this note, it is shown that the Toeplitz operator Tv is hyponormal if and only if g = c + T^f for some constant c and some function h in H°°(D) with «All«, < I- The basis of the proof is a dilation theorem; we will use the notation and formulation of Sarason [13, Theorem 1]. The unilateral (forward) shift on H2 will be denoted by U. Moreover, the proof uses standard results about Hankel operators, for example, see [12] . For rp in L°°, the Hankel operator H^ is the operator on H2
given by H^u = J(I-P)(^u)
where J is the unitary operator from H2± onto H2
Denoting by v* the function v*(el6) = v(e~ie), another way to put this is that Hî s the operator on H2 defined by
(1) (zuv,^) = (H^u,v*), for allv € H°°.

ABSTRACT. For <p in L°°(dD), let <p = f + g where / and g are in H2.
In this note, it is shown that the Toeplitz operator Tv is hyponormal if and only if g = c + T^f for some constant c and some function h in H°°(D) with «All«, < IFor <p in L°°((5D), the Toeplitz operator T^ is the operator on H2 of the unit disk D given by T^u = Ppu where P is the orthogonal projection of L2(dO) onto H2. An operator A is called hypnormal if its self-commutator A* A -AA* is positive. The goal of this paper is to characterize hyponormal Toeplitz operators.
Brown and Halmos began the systematic study of the algebraic properties of Toeplitz operators and showed, [3, p. 98] , that Tv is normal if and only if <p = a+ßp where a and ß are complex numbers and p is a real valued function in L°°. There are many results concerning hyponormality of Toeplitz operators in the literature and properties of hyponormal Toeplitz operators have played an important role in work on Halmos's Problem 5, [7] , "Is every subnormal Toeplitz operator either normal or analytic?" but a characterization has been lacking. (For references, see the bibliography; [6] The basis of the proof is a dilation theorem; we will use the notation and formulation of Sarason [13, Theorem 1] . The unilateral (forward) shift on H2 will be denoted by U. Moreover, the proof uses standard results about Hankel operators, for example, see [12] . For rp in L°°, the Hankel operator H^ is the operator on H2
given by H^u = J(I-P)(^u)
where J is the unitary operator from H2± onto H2
Denoting by v* the function v*(el6) = v(e~ie), another way to put this is that Hî s the operator on H2 defined by • i/vi = Hl>2 if and onlyif {I -P)4>i = (!-P)*l>2-
• H^ll = inf{Halloo : (/ -P)*P = {I-P)<P}-
• Hf = Ä0».
• Either H^ is one-to-one or ker(H^) = xH2 where x is an inner function.
The closure of the range of H^ is H2 in the former case and (x'H2)1-in the latter.
• HtV = \J*H^.
PROOF. Let ip = f + g where / and g are in H2. The first step of the proof is one of the equivalences of Proposition 11 of [6] . For every polyomial p in H2,
Since the polynomials are dense in H2 and since the Hankel and Toeplitz operators involved are bounded, we see that T^ is hyponormal if and only if for all u in H2, (2) \\H-gu\\ < \\Hf-u\\.
Let K denote the closure of the range of Hi, and let S denote the compression of U to K. Since K is invariant for U*, the operator S* is the restriction of U* to K. If HjU\ = Hju2, so that Hj(u\ -u2) =0, then the inequality (2) implies that Hg(u\ -u2) = 0 too and it follows that A is well defined. Moreover, inequality (2) implies ||A|| < 1 so A has an extension to K, which will also be denoted A, with the same norm. Now by the intertwining formula for Hankel operators and the fact that K is invariant for U*, we have HgU = AHf-U = AU'Hj = AS*Hfand also HgU = U*H-g = U*AHf-= S*AHj.
Since the range of Hj is dense in K, we find that AS* = S*A on K, or taking adjoints, that SA* =A*S. H-g = r^-.
Conversely, if equation (3) holds for some k in H°°{D) with ||A||oo < 1> then clearly inequality (2) holds for all u, and T^, is hyponormal.
The proof will be completed by analyzing the relationship given by equation (3) . Using the formulation (1), equation (3) Since the closed span of {zuv: u,v E H°°} is zH2 this means that equation (3) holds if and only if g = c + T~hf for h = k*. (Note that PU«, = ||A*ft«j-) □ In the cases for which Tv is normal, h is a constant of modulus 1 and in the cases for which Tv is known to be subnormal but not normal, h is a constant of modulus less than 1.
It is of some interest to investigate the uniqueness of the functions h that relate / and o. Suppose h\ and h2 are in H°° and C\ + Tj¡-f = g = c2 + Tj-f. This is possible if and only if T-zTirJ = T-zTirJ, that is, if and only if Tzh _zh f = 0. Thus, / must be in (zxH2)x where x is the inner factor of hi-h2. If / is not in any such subspace, the corresponding function h must be unique for every g. On the other hand, if x is an inner function such that / is in (zxH2)1-and ci + Tj-/ = g, then for any /i3 in H°° and h2 = h\ + zxhz, it follows that g = c2 + T^-f for some constant c2. In [6] , the author made the following generalization of the set of g in H2 for
which Tf+g is hyponormal. DEFINITION. Let ¿f = {v E H°° : v(Q) = 0 and ||v||2 < 1}. For / in H2, let Gf denote the set of g in H2 such that for every u in H2,
To see how this definition is relevant to our work, note that if / is in H°° and u is in H2, then by equation (1) (1) / is in G f.
(2) If g is in Gf, then g + \ is in Gf for all complex numbers A. (3) Gf is balanced and convex, that is, if gi and g2 are in Gf and |si| + |s21 < 1, then sigi + s2g2 is also in Gf.
(4) Gf is weakly closed.
(5) T^Gf C Gf for every inner function xConversely, if G is a set that satisfies properties (1) to (5), then G D Gf.
PROOF. That Gf has the indicated properties is Theorem 12 of [6] . To prove the converse statement, note that / is in G and by (3), (4), and (5), G contains T^f whenever h is in the weakly closed convex hull of the set of inner functions. By a theorem of Marshall [11, Corollary, p. 496] , the norm closed convex hull of the Blaschke products in H°° is the unit ball of H°°. Property (2) and Theorem 2 now imply the desired inclusion. D

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