Wolff's Theorem on Ideals for Matrices

arXiv:1304.2918v1 [math.FA] 9 Apr 2013

WOLFF’S THEOREM ON IDEALS FOR MATRICES CALEB D. HOLLOWAY AND TAVAN T. TRENT

Abstract. We extend Wolff’s theorem concerning ideals on H ∞ (D) to the matrix case, giving conditions under which an H ∞ -solution G to the equation F G = H exists for all z ∈ D, where F is an m × ∞ matrix of functions in H ∞ (D), and H is an m × 1 vector of such functions. We then examine several useful results.

1. Introduction In 1962, Lennart Carleson [3] solved the Corona Problem, proving that the ideal I generated by a finite set of functions {fi }ni=1 ⊂ H ∞ (D) is the entire space H ∞ (D) provided there exists δ > 0 such that n X

(1)

i=1

|fi (z)| ≥ δ ∀ z ∈ D.

This result can be extended to hold for infinitely many functions {fi }∞ i=1 (see [8], [13]). Two different extensions of the corona theorem to matrices were given by Fuhrmann [5] and Andersson [1]. Fuhrmann’s result was extended to one-sided infinite matrices by Vasyunin (see Nikolski [7]). However, Treil [10] showed that a complete extension of the corona theorem is not possible in the two-sided infinite matrix case. Trent and Zhang proved that the result of Furmann and Vasyunin can be extended to any algebra that satisfies a corona theorem [15], and later did the same for Andersson’s result, also allowing for one-sided infinite matrices [16]. A more general question than Carleson’s is, under what conditions is a given function h ∈ H ∞ (D) to be found in I? One might suppose based on Carleson’s result that a sufficient condition would be that n X |fi (z)| ≥ |h(z)| ∀ z ∈ D, (2) i=1

but that is not the case, as Rao proved (see Garnett [6]). Thomas Wolff, however, proved that, given (2), h3 ∈ I [17]. More recently, Treil showed that the result fails when the exponent “3” is replaced with “2” (although it holds for any exponent greater than 2) [11]. If we adjust the hypothesis to (3)

[

n X i=1

3

|fi (z)|2 ] 2 ≥ |h(z)| ∀ z ∈ D,

we obtain h ∈ I. A matter of current interest is how this estimate can be improved, which we will discuss at the end of this paper. For now, (3) will be sufficient for our needs. 1

2

CALEB D. HOLLOWAY AND TAVAN T. TRENT

Since the corona theorem has been applied to matrices, a natural question is whether the same can be done for Wolff’s theorem. The answer is yes, as we will now set forth to prove. First, however, a definition is in order. Definition 1.1. Let B ∈ Mm (C). For 1 ≤ k ≤ m, define X det(Eπ BEπ ) detk (B) = π∈Πk (m)

where Πk (m) denotes the increasing k-tuples of integers in {1, 2, . . . , m}. Here Eπ is the m × m matrix whose ith column is the ith column of the m × m identity matrix if i ∈ π, and is zero otherwise. When taking the determinant of Eπ BEπ in the above definition, we delete those columns and rows consisting of all zeros. Wolff ’s Theorem for Matrices. Let F (z) be an m × ∞ matrix of functions in H ∞ (D) with max{rank F (z) | z ∈ D} = k ≤ m. Let H(z) be an m × 1 vector of functions in H ∞ (D). Suppose 3

(i) [detk (F (z)F (z)∗ )] 2 ≥ |hi (z)| ∀ z ∈ D, i = 1, . . . , m (ii) kMF k = 1 (iii) there exists a function u : D → l2 such that F u = H everywhere on D. Then there exists an ∞ × 1 vector G(z) of functions in H ∞ (D) such that (a) F (z)G(z) = H(z) ∀ z ∈ D, and (b) kMG k < ∞. We base our arguments on those found in Trent and Zhang [16]. The main difference here is that we do not assume a uniform lower boundedness on F , and instead assume that F is bounded by the entries of H. 2. Preliminaries Before giving the proof of Wolff’s Theorem for Matrices, we define and list some properties of “Q-operators.” Proofs of these properties can be found in [15]. We let H ∧ K denote the exterior product between two Hilbert spaces H and K, 2 2 and l(n) = ∧ni=1 l2 . In keeping with this notation, l(0) = C. 2 ∞ Let {ei }i=1 denote the standard basis in l . If In denotes increasing n-tuples of positive integers and if (i1 , i2 , . . . , in ) ∈ In , we let πn = {i1 , i2 , . . . , in } and, abusing notation, we write π ∈ In . If we define eπn = ei1 ∧ ei2 ∧ · · · ∧ ein , then {eπn }πn ∈In 2 is defined to be the standard basis for l(n) . Let H(E) be a reproducing kernel Hilbert space on a set E, and let A = M (H(E)), the multiplier algebra on H(E). Let f j (z) = (v1 (z), v2 (z), . . . ), where ∗ {vn }∞ n=1 ⊂ A, such that f j (z)f j (z) ≤ 1 ∀ z ∈ E. Fix z ∈ E, and for n = 0, 1, . . . define (n)∗ 2 2 → l(n+1) Qj (z) : l(n) by (n)∗ Qj (z)(wn ) = f j (z) ∧ wn , (0)∗

2 where wn ∈ l(n) . Note that Qj

(n)∗ ran Qj (z)

(z) = f j (z). (n+1)∗

(z). Furthermore, equality can be ⊂ ker Qj We observe that ∗ shown if we stipulate that f j (z)f j (z) ≥ δ > 0. (This follows from (5) below.)

WOLFF’S THEOREM ON IDEALS FOR MATRICES

3

By the anti-commutivity of the exterior product, we see that (n)

(4)

(n+1)

Qj Qk

(n)

(n+1)

= −Qk Qj

.

2 , we have Also, for eπn ∈ l(n) (n)∗

Qj

∞ X vp (z)ep ) ∧ eπn , (z)(eπn ) = f j (z) ∧ eπn = ( p=1

(n)∗

so with respect to the standard basis, the entries in Qj some n. Thus

(n) Qj (·)

(z) are 0 or ±vn (z) for

has entries belonging to A with respect to the standard basis. (n)

Assume that there exists a fixed z ∈ E such that fj (z) 6= 0, and let Qn = Qj (z). Then for n = 0, 1, . . . , Q∗n Qn + Qn+1 Q∗n+1 = kak2 Il2n+1 .

(5)

⊥ Finally, if ai = f i (z), then for k = 2, . . . , m, let a′k = Psp{a (ak ). Then 1 ,...,ak−1 } ∗ 2 (m−1) (m−1)∗ . . . Q(1)∗ Q am a1 Q(1) a2 a1 = ka1 k a2 . . . Q am

m Y

j=2

ka′j k2

= det(F (z)F (z)∗ ).

(6)

The last equality is obtained by a straightforward computation, using the fact that a∗ 2 i ai kai k2 is the rank one projection of l onto ai . We obtain a very nice norm estimate in the case where A = H ∞ (D). Let A = (a1 , a2 , . . . ), ai ∈ H ∞ (D), with kMA k < ∞. From (5) we have the following estimate on the operator norms (for fixed z ∈ D): kQ(j) k ≤ kAk.

Now kMQ(j) k = sup kQ(j) (z)k ≤ sup kA(z)k = kMA k. z∈D

(Here j ≥ 1. If j = 0, then Q

(j)

(7)

z∈D

= A, so the result is trivial.) Thus kMQ(j) k ≤ kMA k.

We will need Lemma 1 from [16] if we wish to extend our main theorem to spaces besides H ∞ (D), however. The following lemmas will be used in the proof of our theorem. Their proofs have been omitted but may be found in [16]. Definition 2.1. Let {Xj }n+1 j=1 such that Tjk ∈ B(Xj+1 , Xj ).  T11 T12 . . .  T21 T22 . . .  “det”  .  .. Tn1

Tn2

...

be Banach spaces and let {Tjk }nj,k=1 denote operators Let  T1n X T2n   (−1)sgn(σ) T1i1 T2i2 . . . Tnin ..  = .  Tnn

σ∈P (n) σ={i1 ,...,in }

where the products are given in the order indicated and P (n) denotes the permutations of {1, . . . , n} and sgn(σ) denotes the sign of the permutation of σ.

4


Lemma 2.1.  H1  f1   (1) Q1 “det”   ..   . (p−1) Q1

Hi1 fi 1

(1) Qi1

Hip fi

...

(1) Qip

p

.. .

(p−1)

(p−1)

...

Qi1 (1)

... ...

Qip

(p−1)

= p![H1 f i Qi2 . . . Qip 1

+

p X

        (1)

(l−1)

(l)

(p−1)

(−1)l f 1 Hil Qi1 . . . Qil−1 Qil+1 . . . Qip

].

l=1

Lemma 2.2. Suppose F (·) is m×∞ with F (z)F (z)∗ having maximum rank p < m. Suppose that for some function u : E → l2 , F (z)u(z) = H(z), where H = (h1 , . . . , hm ) with hj ∈ M (H(E)). Then for any π ∈ Π(p+1) (n) with π = (j1 , . . . , jp+1 ), we have for each z ∈ E   . . . Hip+1 Hi1  fi  ... fi 1 p+1     (1) (1) . . . Qip+1  = 0. Qi1 “det”    .. ..     . . (p−1) (p−1) . . . Qip+1 Qi1 3. Proof of Wolff’s Theorem for Matrices We are now ready to prove our theorem. Proof. The solution G that we seek can be written as a sum of vectors G1 , . . . , Gm . We will find the vector G1 here; vectors G2 , . . . , Gm are found similarly. Let k ≤ m. By the same process we used to obtain (6), we see that X 3 (k−1) (k−1)∗ (1)∗ (1) (z) . . . Qi2 (z)f ∗i (z)] 2 ≥| hi (z) | f i (z)Qi2 (z) . . . Qik (z)Qik [ 1

1

π∈Πk (m) π={i1 ,...ik }

for all z ∈ D and i = 1, . . . , m. Using (7),

k Mf MQ(1) . . . MQ(k−1) k< ∞, i1

i2

ik

so by Wolff’s Theorem there exists, for each π ∈ Πk (m), a vector vπ with entries in H ∞ (D) such that X (k−1) (1) f i Qi2 . . . Qik vπ = h1 k! 1

π∈Πk (m) π={i1 ,...ik }

and kMvπ k < ∞. We can rewrite this equation in terms of exterior algebras as     (1)   (k−1) f1 Q1 Q1    ..   .  ..    · v1 = h1  . ∧ .  ..  ∧ · · · ∧   fm

(1)

Qm

(k−1)

Qm

WOLFF’S THEOREM ON IDEALS FOR MATRICES

where v1 is a vector with ∞. We claim the vector    G1 = k  

m k

1 0 .. .



0

5

entries vπ for π ∈ Πk (m). Then we have kMvi k
1. 4.4. When H Is a Matrix. Our theorem extends easily to the case where H is an m×n matrix. If F is m×∞, we seek an ∞×n matrix G such that F G = H. Wolff’s Theorem for Matrices allows us to find G by finding its n columns g1 , . . . , gn . What if we wish to solve an equation involving two (or more) matrices F1 and F2 ? That is, we wish to find G1 and G2 such that F1 G1 + F2 G2 = H. This is handled easily if we define F = [F1 F2 ]; that is, F is obtained by concatenating F1 with F2 (rearranging the entries in the case where F1 and F2 are m × ∞). Then, provided hypotheses (i), (ii), and (iii) of our main theorem hold on F , there exists G such that F G = H, and from G we obtain G1 and G2 such that F G = F1 G1 + F2 G2 . References [1] M. Andersson, The corona theorem for matrices, Math. Z. 201 (1989), 121-130 [2] D. Banjade, Wolff ’s problem of ideals in the multiplier algebra on Dirichlet space, to appear [3] L. Carleson, Interpolation by bounded analytic functions and the corona problem, Annals of Math. 76 (1962), 547-559 [4] U. Cegrell, Generalizations of the corona theorem in the unit disc, Proc. Royal Irish Acad. 94 (1994), 25-30 [5] P. A. Fuhrmann, On the corona theorem and its applications to spectral problems in Hilbert space, Trans. Amer. Math. Soc. 132 (1968), 55-66 [6] J. B. Garnett, Bounded Analytic Functions, Academic Press, New York (1981) [7] N. K. Nikolski, Treatise on the Shift Operator, Springer-Verlag, New York (1985) [8] M. Rosenblum, A corona theorem for countably many functions, Integral Equa. Oper. Theory 3 (1980), no. 1, 125-137

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[9] J. Ryle and T. Trent, A corona theorem for certain subalgebras of H ∞ (D), Houston J. Math. 37 (2011), no. 4, 1143-1156 [10] S. R. Treil, Angles between coinvariant subspaces and an operator-valued corona problem, a question of SzoKefalvi-Nagy, Soviet Math. Dokl. 38 (1989), 394-399 [11] S. R. Treil, Estimates in the corona theorem and ideals of H ∞ : A problem of T. Wolff, J. Anal. Math 87 (2002), 481-495 [12] S. R. Treil, The problem of ideals of H ∞ : beyond the exponent 3/2, J. Funct. Anal. 253 (2007), 220-240 [13] T. Trent, Function theory problems and operator theory, Proceedings of the Topology and Geometry Research Center, TGRC-KOSEF, Vol. 8, Dec. 1997 [14] T. Trent, An estimate for ideals in H ∞ (D), Integr. Equat. Oper. Th. 53 (2005), 573-587 [15] T. Trent and X. Zhang, A matricial Corona Theorem, Proc. Amer. Math. Soc. 134 (2006), 2549-2558 [16] T. Trent and X. Zhang, A matricial corona theorem II, Proc. Amer. Math. Soc. 135 (2007), 2845-2854 [17] T. Wolff, A refinement of the corona theorem, in Linear and Complex Analysis Problem Book, by V. P. Havin, S. V. Hruscev, and N. K. Nikolski (eds.), Springer-Verlag, Berlin (1984)