J Geom Anal DOI 10.1007/s12220-016-9740-9

Nehari-Type Theorem for Non-commutative Hardy Spaces F. Sukochev1 · K. Tulenov2,3 · D. Zanin1

Received: 5 July 2016 © Mathematica Josephina, Inc. 2016

Abstract In this paper, we give an answer to a conjecture due to Muscalu. We also prove a non-commutative analogue of Cwikel’s theorem. Keywords Von Neumann algebra · Non-commutative Hardy space · Non-commutative symmetric space · Nehari-type theorem Mathematics Subject Classification 46L51 · 46L52

1 Introduction and Preliminaries In this paper, we resolve an open question due to Muscalu [15, p. 691] in the affirmative. Let L p for 0 < p ≤ ∞ denote the space of absolutely p-integrable functions on R equipped with Lebesgue measure, and let H p be the corresponding Hardy subspace. In his work [15], Muscalu resolved affirmatively the following question:

F. Sukochev and D. Zanin research is supported by the ARC.

B

F. Sukochev [email protected] K. Tulenov [email protected] D. Zanin [email protected]

1

School of Mathematics and Statistics, University of New South Wales, Kensington, NSW 2052, Australia

2

Institute of Mathematics and Mathematical Modeling, 050010 Almaty, Kazakhstan

3

Al-Farabi Kazakh National University, 050040 Almaty, Kazakhstan

123

F. Sukochev et al.

Question 1.1 Is there any  constant C(= C( p1 , m)) > 0 such that for every f ∈ m m  L there exists f ∈ p k k=1 k=1 H pk satisfying the estimate f −  f  L pk ≤ C · dist L pk ( f, H pk ) for every 1 ≤ k ≤ m? In the same paper, he also conjectured that the non-commutative version of Question 1.1 also has an affirmative resolution. In this paper, we confirm this conjecture. We need first some notions and notations from the theory of symmetric operator spaces and non-commutative integration theory (see, e.g. [5,12,17,21]). 1.1 Quasi-Banach Symmetric Function Spaces Let L(0, ∞) be the spaces of all measurable real-valued functions on (0, ∞) equipped with Lebesgue measure m (functions which coincide almost everywhere are considered identical). Define S(0, ∞) to be the subset of L(0, ∞) which consists of all functions x such that m({t : |x(t)| > s}) is finite for some s > 0. For x ∈ S(0, ∞), we denote by μ(x) the decreasing rearrangement of the function | f |. That is, μ(t, x) = inf{s ≥ 0 : m({|x| > s}) ≤ t}, t > 0.

Definition 1.2 We say that (E,  ·  E ) is a symmetric quasi-Banach function space if the following holds. (a) E is a subset of S(0, ∞). (b) (E,  ·  E ) is a quasi-Banach space. (c) If x ∈ E and if y ∈ S(0, ∞) are such that |y| ≤ |x|, then y ∈ E and y E ≤ x E . (d) If x ∈ E and if y ∈ S(0, ∞) are such that μ(y) = μ(x), then y ∈ E and y E = x E . Furthermore, we recall that the quasi-norm in E is said to be order continuous if, for every sequence {xn }n≥0 ⊂ E such that xn ↓ 0 in S(0, ∞), we have that  f n  E → 0. Order continuity of the quasi-norm is equivalent to separability of the space E (see [11]). Special examples of such quasi-Banach function spaces are the spaces L p (0, ∞), 0 < p ≤ ∞, equipped with their usual quasi-norm  ·  p . We recall that every symmetric Banach function space satisfies (L 1 ∩ L ∞ )(0, ∞) ⊂ E ⊂ (L 1 + L ∞ )(0, ∞) with continuous embeddings. For more details, see [10].

123

Nehari-Type Theorem for Non-commutative Hardy Spaces

We say that y is submajorised by x in the sense of Hardy–Littlewood (written y ≺≺ x) if 

t 0



t

μ(s, y)ds ≤

μ(s, x)ds, t > 0.

0

For brevity, we introduce the following definition. Definition 1.3 Let p ∈ (0, ∞). A symmetric quasi-Banach function space (E,  ·  E ) on (0, ∞) is said to be a p-space if for all x ∈ E and y ∈ S(0, ∞), (a) μ p (y) ≺≺ μ p (x) and x ∈ E implies that y ∈ E and y E ≤ x E . p p p (b) x + y E ≤ x E + y E for all x, y ∈ E. Evidently, L p is a p-space for every 0 < p ≤ 1. Observe that it is implicit in the definition that every p-space E is a subset of (L p + L ∞ )(0, ∞). 1.2 Quasi-Banach Dymmetric Operator Spaces Let M be a von Neumann algebra on a separable Hilbert space H equipped with a semi-finite normal faithful trace τ. The commutant of M is denoted by M . A closed densely defined linear operator T in H is said to be affiliated with M if it commutes with the algebra M . An operator T affiliated with M is said to be τ -measurable if for every  > 0, there exists a projection P ∈ M such that P(H ) ⊂ dom(T ) and τ (1 − P) < . The set of all τ -measurable operators will be denoted by S(M, τ ). The set S(M, τ ) is a ∗-algebra with sum and product being closures of the algebraic sum and product (see [12]). A self-adjoint operator is affiliated with M if and only if its spectral projections are in M. We have T ∈ S(M, τ ) if and only if   τ E |T | (t, ∞) < ∞ for some t > 0. Recall the definition of the generalised singular value function of an operator T ∈ S(M, τ ). For s > 0     μ(s, T ) = inf t > 0 : τ E |T | (t, ∞) ≤ s , s > 0. For more details on generalised singular value functions, we refer the reader to [5] and [12]. Recall the construction of a quasi-Banach symmetric operator space E(M, τ ). Let E be a quasi-Banach symmetric function space. Set   E(M, τ ) = T ∈ S(M, τ ) : μ(T ) ∈ E . We equip E(M, τ ) with a natural quasi-norm T  E(M,τ ) = μ(T ) E , T ∈ E(M, τ ).

123

F. Sukochev et al.

The following fundamental questions were resolved in [8] (see also [12]) in the affirmative: Question 1.4 Suppose E is a symmetric Banach function space. (a) Is it true that E(M, τ ) is normed? (b) Is it true that E(M, τ ) is Banach? It was further established in [21] that E(M, τ ) is quasi-Banach. 1.3 Noncommutative Hardy Spaces Let N be a complete nest of projections in M (see [4]). We say that T ∈ S(M, τ ) is triangular if every projection P ∈ N is T -invariant (i.e. T P = P T P). Define the non-commutative Hardy space associated with E(M, τ ) and the nest N as follows (cf.[18]). Definition 1.5 A Hardy space H E (M, τ ; N ) is the set of all operators in E(M, τ ) which are triangular with respect to N . We need to recall some basic notions from classical interpolation theory (cf. [2]). Let (X 0 , X 1 ) be a compatible couple of quasi-Banach spaces. This means that X 0 and X 1 are continuously embedded into a larger topological vector space. Define the Peetre K -functional as follows: for all x ∈ X 0 + X 1 and for all t > 0, set  K (t, x, X 0 , X 1 ) := inf x0  X 0 + tx1  X 1 :

 x = x0 + x1 , x0 ∈ X 0 , x1 ∈ X 1 .

Let Y0 ⊂ X 0 and Y1 ⊂ X 1 be closed subspaces. We say that the compatible couple (Y0 , Y1 ) is K -closed in (X 0 , X 1 ) if, for every y ∈ Y0 + Y1 , we have K (t, y, Y0 , Y1 ) ≤ C · K (t, x, X 0 , X 1 ), t > 0. Here, the constant C is independent of y and t. Observe that the converse inequality is immediate. Perhaps the most interesting and general results regarding Nehari-type theorems for Hardy spaces are obtained in [7,15]. It is well known that if f ∈ L ∞ , f ∞ = dist∞ ( f, H∞ ) (cf. [20]). then there exists  f ∈ H∞ such that  f −  In [7, Theorem 6], Kaftal et al. proved the following Nehari-type result: For every φ ∈ L ∞ and δ > 1, there exists ψ ∈ H∞ such that φ − ψ∞ ≤ δd(φ, H∞ ) and φ − ψ2 ≤ δ(δ 2 − 1)−1/2 d2 (φ, H∞ ) where d2 (φ, H∞ ) denotes the L 2 -distance of φ from H∞ . This result was further extended by Muscalu [15, Theorem 2.4]. In [7], those authors have used this result to obtain another proof of Sarason’s theorem on the closure of H∞ + C, where C is a linear space of continuous functions [20]. It was observed by Pisier in [16] that [7, Theorem 6] can be rephrased in terms of the K -closedness of the couple (H2 , H∞ ) in (L 2 , L ∞ ).

123

Nehari-Type Theorem for Non-commutative Hardy Spaces

The main task of the first part of the present work is to give a non-commutative analogue of a theorem due to Cwikel originally established in [3] (see also [2, Theorem 3.4]) for commutative H p -spaces. The proof of a non-commutative analogy of Cwikel’s theorem will be presented in Sect. 2. In Sect. 3, we prove a Nehari-type theorem for m-tuples of non-commutative Hardy spaces. Our results extend those in [7,16] and are more precise than the corresponding statement in [6] even in the case of Schatten classes. We also contribute to the resolution of [7, Question (v), p. 400]. Our Theorem 3.3 answers an open question stated in [15]. Our main instrument is a very recent development in the theory of non-commutative H p -spaces associated with semi-finite subdiagonal algebra [1] due to Bekjan, who extended results for various special cases due to Pisier [16], Marsalli and West [13,14] and Randrianantoanina [19] to the general setting. We use results from [1] in the proof of crucial Lemma 2.2.

2 Cwikel’s Theorem for Non-commutative H p -Spaces For brevity, in this section we set L p = L p (M, τ ) and H p = H p (M, τ ; N ). Lemma 2.1 If T1 , T2 ∈ (L p + L ∞ )(M, τ ), 0 < p ≤ 1, then μ p (T1 + T2 ) ≺≺ μ p (T1 ) + μ p (T2 ). Proof Suppose first that T1 and T2 are both positive and τ -compact, i.e. μ(∞, T j ) = 0, j = 1, 2. Recall that 

t

μ(s, T )ds = sup {τ (P T P) : P T = T P, τ (P) ≤ t}

(2.1)

0

for every positive τ -compact operator T (see [5, Lemma 4.1]). By [5, Lemma 2.5 (iv)] and (2.1) we have 

t 0

μ p (s, T1 + T2 )ds = sup τ (P(T1 + T2 ) p P) : (T1 + T2 )P

= P(T1 + T2 ), τ (P) ≤ t = sup τ ((P(T1 + T2 )P) p ) : (T1 + T2 )P

= P(T1 + T2 ), τ (P) ≤ t p

≤ sup{P(T1 + T2 )P p :

τ (P) ≤ t}.

On the other hand, by in [5, Theorem 4.9 (iii)] we have p

p

p

P(T1 + T2 )P p ≤ P T1 P p + P T2 P p , it follows that

123

F. Sukochev et al.

 0

t

p

p

μ p (s, T1 + T2 )ds ≤ sup{P T1 P p , τ (P) ≤ t} + sup{P T2 P p , τ (P) ≤ t}  t  t ≤ μ p (s, T1 )ds + μ p (s, T2 )ds. 0

0

This proves the assertion for the case when T1 and T2 are both positive and τ -compact. Suppose now that T1 and T2 are positive. Fix a sequence Pk ↑ 1 of τ -finite projections. We have μ(Pk T1 Pk ) ↑ μ(T1 ), μ(Pk T2 Pk ) ↑ μ(T2 ), μ(Pk (T1 + T2 )Pk ) ↑ μ(T1 + T2 ), k ↑ ∞, (see the proof of [5, Proposition 2.7]). It follows from the preceding paragraph and the Monotone Convergence Principle that μ p (T1 + T2 ) ≺≺ μ p (T1 ) + μ p (T2 ). for the case when T1 and T2 are both positive. Consider now the most general case. Let T1 , T2 ∈ (L p + L ∞ )(M, τ ). Then, there exist partial isometries U and V such that (see [5, Lemma 4.3]) |T1 + T2 | ≤ U |T1 |U ∗ + V |T2 |V ∗ . We have μ p (T1 + T2 ) ≤ μ p (U |T1 |U ∗ + V |T2 |V ∗ ) ≺≺ μ p (U |T1 |U ∗ ) + μ p (V |T2 |V ∗ ) ≤ μ p (T1 ) + μ p (T2 ).  

This completes the proof.

The following assertion was proved in [16] for p ≥ 1. To prove it for every p > 0, we use the results of [1,9]. The notation A ≈ p B means that there exist constants (1) (2) C p , C p (depending on p) such that (2) C (1) p A ≤ B ≤ C p A.

Lemma 2.2 If T ∈ HL p +L ∞ (M, τ ; N ), 0 < p < 1, then  K (t, T, H p , H∞ ) ≈ p

tp

1

p

μ p (s, T )ds

, t > 0.

(2.2)

0

Proof According to the Holmstedt formula in [2, p. 307], the assertion can be rewritten as follows. K (t, T, H p , H∞ ) ≈ p K (t, T, L p , L ∞ ), t > 0.

123

Nehari-Type Theorem for Non-commutative Hardy Spaces

That is, the equivalence (2.2) asserts the K -closedness of the couple (H p , H∞ ) in the couple (L p , L ∞ ). Let us fix 1 < s < u < ∞ so that s > p; additionally let θ = 1− sp and δ = 1− up , so that θ < δ. We set X 0 = L p , X 1 = L ∞ , E 0 = L s and E 1 = L u . Similarly, let Y0 = H p , Y1 = H∞ , F0 = Hs and F1 = Hu . Clearly, by [22, Theorem 4.1] we have E 0 = (X 0 , X 1 )θ,s ,

E 1 = (X 0 , X 1 )δ,u .

where (·, ·)θ,s -real interpolation (see [2, Definition 1.7, p. 299]) and by [1, Theorem 6.3], we also have F0 = (Y0 , Y1 )θ,s ,

F1 = (Y0 , Y1 )δ,u .

[9, Theorem 1.2] states that “If the couple (Y0 , F1 ) is K -closed in (X 0 , E 1 ) and the couple (F0 , Y1 ) is K -closed in (E 0 , X 1 ), then the couple (Y0 , Y1 ) is K -closed in (X 0 , X 1 )”. We now verify the conditions of the just cited theorem. That (Y0 , F1 ) = (H p , Hu ) is K -closed in (X 0 , E 1 ) = (L p , L u ) is established in [1, Lemma 6.5 (i)]. That (F0 , Y1 ) = (Hs , H∞ ) is K -closed in (E 0 , X 1 ) = (L s , L ∞ ) is established in [1, Lemma 6.5 (iii)]. Thus, [9, Theorem 1.2] is applicable and we obtain that the couple (Y0 , Y1 ) =   (H p , H∞ ) is K -closed in the couple (X 0 , X 1 ) = (L p , L ∞ ). The following theorem is a non-commutative analogue of Cwikel’s theorem [2, Theorem 3.4.] which is the main result of this section. Theorem 2.3 If T ∈ HL p +L ∞ (M, τ ; N ), 0 < p < 1, then there exists a decomposition T =

Sk , Sk ∈ HL p ∩L ∞ (M, τ ; N )

k∈Z

such that 

t

μ p (s, T )ds ≈ p

0

p p

min Sk  p , tSk ∞ , t > 0.

k∈Z

Proof Assume that T ∈ / L p (M, τ ) and T ∈ / L ∞ (M, τ ). For the cases, when T ∈ L p (M, τ ) or T ∈ L ∞ (M, τ ) the proof is similar and easier. Indeed, if T ∈ L p (M, τ ) or T ∈ L ∞ (M, τ ), then the summing in the proof below is taken over Z− or Z+ , respectively. In particular, the summing is taken over finite set of indexes, if T ∈ (L p ∩ L ∞ )(M, τ ). For a given t > 0, set ξ(t) to be the least number such that  0

ξ(t)



t

μ (s, T )ds ≥ 2 p

0

μ (s, T )ds, p

1 ξ(t)



ξ(t) 0

1 μ (s, T )ds ≤ 2t



p

t

μ p (s, T )ds.

0

123

F. Sukochev et al.

Such ξ(t) exists due to the assumptions. Moreover, the mapping t → ξ(t) is a bijection from (0, ∞) onto itself. Define an increasing sequence {t j } j∈Z by setting t0 = 0 and t j+1 = ξ(t j ) for every j ∈ Z. It follows from Lemma 2.2, that, for every j ∈ Z, there exists a decomposition T = A j + B j (both A j and B j are triangular) such that 

p

A j  p ≤ C p

tj

μ p (s, T )ds,

j ∈ Z,

(2.3)

0 p t j B j ∞



tj

≤ Cp

μ p (s, T )ds,

j ∈ Z.

(2.4)

0

Set S j = A j+1 − A j . Step 1 We show that ∞

T =

Sk ,

k=−∞

where the series converges in (L p + L ∞ )(M, τ ). Clearly, T−

j−1

Sk = T − A j + A− j = B j + A− j .

k=− j

It follows from the definition of a sequence {t j } j∈Z and (2.3) that A− j  p → 0 as j → ∞. It follows from the definition of a sequence {t j } j∈Z and (2.4) that B j ∞ → 0 as j → ∞. Thus, T −

j−1

Sk  L p +L ∞ ≤ B j ∞ + A− j  p → 0,

j → ∞.

k=− j

This proves Step 1. Step 2 We show that

p



p

min{Sk  p , tSk ∞ } ≤ 18C p

k∈Z

t

μ p (s, T )ds.

0

For a given t > 0, choose k0 such that t ∈ [tk0 , tk0 +1 ]. We have

k
123

p

Sk  p ≤

k
p

p (2.3)

Ak  p + Ak+1  p ≤ 2C p

 k≤k0

0

tk

μ p (s, T )ds.

Nehari-Type Theorem for Non-commutative Hardy Spaces

It follows from the definition of a sequence {t j } j∈Z that



p Sk  p

tk 0

≤ 4C p



0

k
t

μ (s, T )ds ≤ 4C p p

μ p (s, T )ds.

(2.5)

0

Similarly, we have

p tSk ∞

≤2 t p

k>k0

p Bk ∞

1  tk 4C p t μ p (s, T )ds. tk 0

p (2.4) + Bk+1 ∞ ≤

k>k0

k>k0

It follows now from the definition of a sequence {t j } j∈Z that

k>k0



1

p

tSk ∞ ≤ 8C p t

tk0 +1

tk0 +1



t

μ p (s, T )ds ≤ 8C p

μ p (s, T )ds.

(2.6)

0

0

It remains to consider the summand with k = k0 . We either have 

tk0 +1

 μ p (s, T )ds = 2

0

tk 0

μ p (s, T )ds

0

or 

1 tk0 +1

tk0 +1

0

1 μ (s, T )ds = 2tk0



tk 0

p

μ p (s, T )ds.

0

In the first case, we have p

p



p

Sk0  p ≤ Ak0  p + Ak0 +1  p ≤ 4C p

tk 0

μ p (s, T )ds.

0

In the second case, we have p tSk0 ∞

≤2

p

p t (Bk0 ∞



p + Bk0 +1 ∞ )

t

≤6

μ p (s, T )ds.

0

In either case, we have p

p



min{Sk0  p , tSk0 ∞ } ≤ 6C p

t

μ p (s, T )ds.

(2.7)

0

Combining (2.5), (2.6) and (2.7), we infer the claim of Step 2. Step 3 We show that  0

t

μ p (s, T )ds ≤

p

p

min{Sk  p , tSk ∞ }.

(2.8)

k∈Z

123

F. Sukochev et al.

It follows from Lemma 2.1. that 

t



μ p (s, T )ds ≤

0

k∈Z

t

μ p (s, Sk )ds.

0

Clearly, 

t



0



t

∞

μ p (s, Sk )ds ≤ 

t

μ p (s, Sk )ds ≤

0

p

μ p (s, Sk )ds = Sk  p ,

0 p

μ p (0, Sk )ds = tSk ∞ .

0

This proves the claim of Step 3. Combining the assertions of Steps 1, 2 and 3, we complete the proof.

 

3 Nehari-Type Theorem for Non-commutative Hardy Spaces The following is our key lemma: Lemma 3.1 Let T ∈ HL p +L ∞ (M, τ ; N ), 0 < p < 1. If T = T1 + T2 , where T1 , T2 ∈ (L p + L ∞ )(M, τ ), then there exists a decomposition T = A1 + A2 , where A1 , A2 ∈ HL p +L ∞ (M, τ ; N ) and μ p (A j ) ≺≺ C p μ p (T j ),

j = 1, 2.

Proof Firstly, let T =

Sk

k∈Z

be a decomposition given in Theorem 2.3. Set ψ=

p

Sk ∞ χ

k∈Z

0,

p  Sk  p p Sk ∞

.

Clearly, ψ = μ(ψ). By Theorem 2.3, we also have 

t

μ p (s, T )ds ≈ p

0

=

 t 0 k∈Z

123

p

p

min{Sk  p , tSk ∞ }

k∈Z p

Sk ∞ χ

p  S  0, k pp Sk ∞



t

(s)ds = 0

μ(s, ψ)ds,

(3.1)

Nehari-Type Theorem for Non-commutative Hardy Spaces

For simplicity, given a sequence a ∈ ∞ (Z), we use a shorthand for the formal infinite sum,

a(k)χ Sk  pp  . (3.2) L(a) = 0,

k∈Z

p Sk ∞

Since T = T1 + T2 , it follows from the equivalence above (3.1) and Lemma 2.1 that  t   t  t  t (2) p (2) p p μ(s, ψ)ds ≤ C p μ (s, T )ds ≤ C p μ (s, T1 )ds + μ (s, T2 )ds . 0

0

0

0

By [2, Theorem III 7.7], there exist positive sequences θ1 , θ2 ∈ ∞ (Z) such that θ1 + θ2 = 1 and such that L(θ j ) · ψ ≺≺ C p μ p (T j ),

j = 1, 2.

(3.3)

Let ξ : [0, 1] → R be such that (1) ξ(t) + ξ(1 − t) = 1 for every t ∈ [0, 1]. (2) ξ(t) = t 1/ p , t ∈ [0, 14 ].   −2 (3) ξ(t) ∈ (2 p , 21 ) for t ∈ 41 , 21 . Clearly, ξ p (t) ≤ 22− p t, t ∈ [0, 1]. We set Aj =

ξ(θ j (k))S j .

k∈Z

We have 

t

L .2.1

μ p (s, A j )ds ≤

0



k∈Z (3.1)

≤ 22− p

(3.2)

t

ξ p (θ j (k))

= 22− p

μ p (s, S j )ds

0

p p

θ j (k) min Sk  p , tSk ∞

k∈Z t



μ(s, L(θ j ) · ψ)ds  t (3.3) ≤ 22− p C p μ p (s, T j )ds. 0

0

  Remark 3.2 If E is a p-space, then the concavity modulus of E(M, τ ) does not exceed 1

2 p (cf. [11] for more details on the concavity modulus). Proof Indeed, we have μ(T1 + T2 ) ≤ σ2 μ(T1 ) + σ2 μ(T2 ),

123

F. Sukochev et al.

where the dilation operator σ2 acting on the L(0, ∞) is defined by setting (cf. [12, p.85])   t , x ∈ L(0, ∞), t > 0. σ2 (x)(t) := x 2 Observe that σ2 μ(T ) is equimeasurable with the function f + g, f, g ∈ (L 1 + L ∞ )(0, ∞), where μ( f ) = μ( f ) = μ(T ) and f g = 0. Hence, it follows from the definition of a p-space that p

p

p

σ2 μ(T ) E(M,τ ) =  f + g E(0,∞) ≤ 2T  E(M,τ ) . Again using definition of a p-space, we infer p

p

p

p

T1 + T2  E(M,τ ) ≤ σ2 μ(T1 ) + σ2 μ(T2 ) E ≤ 2T1  E(M,τ ) + 2T2  E(M,τ ) .   The following theorem is the main result of this section which is an extension of [7, Theorem 6] (see also [15, Theorem 1.1]) and completely answers Question 1.1 in the non-commutative case. m Theorem 3.3 Let {E k }m k=1 E k (M, τ ) there exists k=1 be p-spaces. For every T ∈ S ∈ H∩mk=1 E k (M, τ ; N ) such that T − S E k ≤ cmp · dist E k (T, H E k ), 1 ≤ k ≤ m. Proof Let C p be the constant in Lemma 3.1. Without loss of generality, C p ≥ 4. We 2

set c p = C pp . We prove the assertion by induction on m. For m = 1, the assertion is trivial because c p ≥ 1. Suppose it is true for m ≥ 1. Let us prove it for m + 1. By induction, there is S1 ∈ H∩mk=1 E k (M, τ ; N ) such that T − S1  E k ≤ cmp · dist E k (T, H E k ), 1 ≤ k ≤ m. Trivially, there exists S2 ∈ H E m+1 (M, τ ; N ) such that T − S2  E m+1 ≤ 2 · dist E m+1 (T, H E m+1 ). Set B1 = T − S1 and B2 = T − S2 . We then have S1 − S2 = B2 − B1 . Since every pspace E is a subset of (L p + L ∞ )(0, ∞) and E k = L pk implies H E k = HL pk , applying Lemma 3.1 to the operator S1 − S2 , we find a decomposition S1 − S2 = A2 − A1 such that A1 , A2 ∈ (H p + H∞ )(M, τ ; N ) and such that μ p (A j ) ≺≺ C p μ p (B j ),

123

j = 1, 2.

Nehari-Type Theorem for Non-commutative Hardy Spaces

Set S = S1 + A1 = S2 + A2 . Then, we have T − S E k = (T − S1 ) − A1  E k 1

1

≤ 2 p T − S1  E k + 2 p A1  E k 1

1

≤ (1 + C pp )2 p T − S1  E k ≤ c p · cmp · dist E k (T, H E k ), 1 ≤ k ≤ m. Also, T − S E m+1 = (T − S2 ) − A2  E m+1 1

1

≤ 2 p T − S2  E m+1 + 2 p A2  E m+1 1

1

≤ (1 + C pp )2 p T − S2  E m+1 ≤ c p · dist E m+1 (T, H E m+1 ). This completes the proof.

 

In the special case of L p -spaces, we obtain the following corollary which is a noncommutative extension of [15, Corollary 2.5], and which answers the question in [15, p. 691]:  Corollary 3.4 Let 0 < p1 < · · · < pm . For every T ∈ m k=1 L pk (M, τ ) there exists S ∈ H∩mk=1 L pk (M, τ ; N ) such that T − S pk ≤ cmp · dist L pk (T, HL pk ), 1 ≤ k ≤ m.

References 1. Bekjan, T.: Noncommutative Hardy space associated with semi-finite subdiagonal algebras. J. Math. Anal. Appl. 429(2), 1347–1369 (2015) 2. Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press, New York (1988) 3. Cwikel, M.: K-divisibility of the K-functional and Calderón couples. Ark. Mat. 22, 39–62 (1984) 4. Davidson, K.: Nest Algebras. Pitmann Research Series (1988) 5. Fack, T., Kosaki, H.: Generalised s-numbers of τ -measurable operators. Pac. J. Math. 123, 269–300 (1986) 6. Janson, S.: Interpolation of subcouples and quotient couples. Ark. Mat. 31(2), 307–338 (1993) 7. Kaftal, V., Larson, D., Weiss, G.: Quasitriangular subalgebras of semifinite von Neumann algebras are closed. J. Funct. Anal. 107, 387–401 (1992) 8. Kalton, N., Sukochev, F.: Symmetric norms and spaces of operators. J. Reine Angew. Math. 621, 81–121 (2008) 9. Kisliakov, S.: Interpolation of H p -spaces: some recent developments. Function spaces, interpolation spaces, and related topics (Haifa 1995), Israel Mathematica and conference proceedings . Bar-Ilan University, Ramat Gan 13, pp. 102–140 (1999) 10. Krein, S., Petunin, Y., Semenov, E.: Interpolation of Linear Operators. American Mathematical Society, Providence (1982) 11. Lindenstrauss, J., Tzafiri, L.: Classical Banach Spaces, II edn. Springer, New York (1979) 12. Lord, S., Sukochev, F., Zanin, D.: Singular Traces. Theory and Applications: De Gruyter Studies in Mathematics. De Gruyter, Berlin (2013)

123

F. Sukochev et al. 13. Marsalli, M., West, G.: Noncommutative H p -spaces. J. Oper. Theory 40, 339C–355C (1997) 14. Marsalli, M., West, G.: The dual of noncommutative H1 ,. Indiana Univ. Math. J. 47, 489C–500 (1998) 15. Muscalu, C.: A joint norm control Nehari type theorem for N -tuples of Hardy spaces. J. Geom. Anal. 9(4), 683–691 (1999) 16. Pisier, G.: Interpolation between H p -spaces and noncommutative generalization. Pac. J. Math. 155(2), 341–368 (1992) 17. Pisier, G., Xu, Q.: Noncommutative L p -spaces. In: Handbook of the Geometry of Banach Spaces. 2, 1459–1517 (2003) 18. Power, S.: Factorization in analytic operator algebras. J. Funct. Anal. 67, 413–432 (1986) 19. Randrianantoanina, N.: Hilbert transform associated with finite maximal subdiagonal algebras. Aust. Math. Soc. 65, 388C–404 (1999) 20. Sarason, D.: Generalised interpolation in H ∞ . Trans. Am. Math. Soc. 127, 179–203 (1967) 21. Sukochev, F.: Completeness of quasi-normed symmetric operator spaces. Indag. Math. 25, 376–388 (2014) 22. Xu, Q.: Applications du théorème de factorisation pour des à fonctions o valeurs opératours. Studia Math. 95, 273–292 (1990)

123