Sequence systems

In the following we will need similar properties of two slightly different types of nuclear operators defined by a triplet of sequence spaces. For this, it is convenient to introduce the following general definition.

Definition 1.41. By a sequence system α we mean a rule that for every Banach space X fixes a linear subspace α(X) of X^N such that for every x = (x_n)∈α(X), for every Banach space Y and for every T ∈ L(X, Y) one has T ◦x= (T x_n)∈α(Y).

Definition 1.42. By a w^∗-sequence system α we mean a rule that for every dual Banach space X^∗ fixes a linear subspace α(X^∗) of (X^∗)^N such that for every x^∗ = (x^∗_n) ∈ α(X^∗), for every Banach space Y and for every T ∈ L(Y, X)one has T^∗ ◦x^∗ = (T^∗x^∗_n)∈α(Y^∗).

Note that every sequence system is, in particular, also a w^∗-sequence system.

Example 1.43. If λ is equipped with a linear topology, then also ˆλ^w is a sequence system and λˆ^w∗ is a w^∗-sequence system. We explain why the first claim holds. Indeed, from x ∈ λ(X) and T ∈ L(X, Y) follows that (T x_n) ∈ λ(Y). Thus for every y^∗ ∈ Y^∗ we have (y^∗(T x_n))_n ∈ λ. Therefore j_w[(T x_n)]∈ L(Y^∗, λ), meaning (T x_n)∈λˆ^w(Y).

Example 1.44. Given a sequence space λ, both λ^w : X 7→ λ^w(X) and λ^s : X 7→λ^s(X)(assuming thatλ is solid) are sequence systems andλ^w∗ :X^∗ 7→

λ^w∗(X^∗) is a w^∗-sequence system. We explain why the claim for λ^s holds.

Indeed, from x∈λ(X)we have (kx_nk)_n∈λ. Sinceλ is solid, it follows from kT x_nk ≤ kTkkx_nk that (kT x_nk)∈λ. Thus, (T x_n)∈λ^s(Y).

LetX be a Banach space and let q be a quasi-norm on a vector sequence space α(X).

Definition 1.45. We say that a sequence system (w^∗-sequence system) α is quasi-normed, if for all Banach spaces X the vector sequence spaceα(X) (α(X^∗)) is equipped with a quasi-norm q such that for every Banach space Y, every operator T ∈ L(X, Y) (R ∈ L(Y, X)) and every x ∈ α(X) (x^∗ ∈ α(X^∗)), one has q(T ◦x)≤ kTkq(x) (q(R^∗◦x^∗)≤ kRkq(x^∗)).

Given any property P introduced for (quasi-normed) vector sequence spaces, we will say that a (quasi-normed) (w^∗-)sequence system α enjoys property P whenever every component of α enjoys P.

We say that a sequence system α is non-zero, when every component of α is non-zero.

Example 1.46. If λ is monotone and q is monotone, then both λˆ^w and λˆ^w∗ are monotone with monotone quasi-norms. We explain why the claim for ˆλ^w holds. Indeed, let λ be monotone, let X be a Banach space, let x ∈λˆ^w(X), N ⊂ N and x^∗ ∈ X^∗. Then x^∗ ◦ S_N(x) = S_N[(x^∗(x_n))] ∈ λ. Since q is monotone, it follows that the the norm of operator jwSN(x) is bounded.

Therefore λˆ^w is monotone. Let q be a monotone quasi-norm on λ. Let x, y ∈X be such that k · k ◦y ≤ k · k ◦x. Since

kxk_w =kj_wxk= sup

kx^∗k≤1

q((j_wx)x^∗) = sup

kx^∗k≤1

q((x^∗(x_n))_n)

and q is monotone, we have that kykw ≤ kxkw and therefore k · kw is also monotone.

Example 1.47. Let λ be a sequence space equipped with a quasi-norm q. If λ is symmetric and q is K-symmetric, then both λˆ^w and λˆ^w∗ are symmetric with K-symmetric quasi-norms.

2 (λ, µ, ν )-nuclear operators

In this section we introduce the notion of (λ, α, β)-nuclear operators. In the following, we assume that sequence spaces and sequence systems are non-zero.

Definition 2.1. Letλbe a sequence space, letαbew^∗-sequence system and letβ be a sequence system. Let X and Y be Banach spaces. We say that an operator T ∈ L(X, Y) is(λ, α, β)-nuclear if

T =

∞

X

n=1

σ_nx^∗_n⊗y_n,

where (σ_n) ∈λ, (x^∗_n) ∈ α(X^∗) and (y_n) ∈β(Y) (the series converges in the usual operator norm). Here, givenx^∗ ∈X^∗ andy∈Y, the one-rank operator x^∗ ⊗y is defined by (x^∗⊗y)(x) = x^∗(x)·y for all x ∈ X. We denote the collection of of all(λ, α, β)-nuclear operators byN_(λ,α,β). In the following, we will also use notation T =P

σx^∗⊗y =P∞

n=1σ_nx^∗_n⊗y_n for a given nuclear representation of operator T.

Given a triplet of sequence spaces λ, µ, ν, we denote N_(λ,µ,ν) :=

N_(λ,µw∗

,ν^w). The operators belonging to the latter collection will be called (λ, µ, ν)-nuclear. Often, we will also consider a related class N_(λ,µ^w_,ν^w₎ with-out giving it a special notation.

Example 2.2. Let 1 ≤ r, p, q ≤ ∞ and 1 + 1/r ≥ 1/p+ 1/q. The (r, p, q)-nuclear operators (see [[Pie80, Chapter 18]]) by definition are exactly the (`<sub>r</sub>, `^w_q∗, `<sup>w</sup><sub>p</sub>∗)-nuclear operators, which in this special case coincide with the (`_r, `<sub>q</sub><sup>∗</sup>, `_p^∗)-nuclear operators, due to Lemma 1.39.

Example 2.3. The λ-nuclear operators (see [Ram70]) are exactly the (λ, `_∞, λ^×)-nuclear operators. Note that Ramanujan assumed thatkx^∗_nk ≤1 and ky_nk ≤ 1 for all n ∈ N. These conditions can be attained by normal-izing the nuclear-representation of the operator, considering σ_nx^∗_n ⊗ y_n = k(x^∗_n)kk(y_n)kσ_nk(x^∗_n)k⁻¹k(y_n)k⁻¹x^∗_n⊗y_n.

We continue with showing that under some modest assumptions,N_(λ,α,β) is an operator ideal.

Theorem 2.4. Let λ be a sequence space, let α be w^∗-sequence system and let β be a sequence system. Let λ, α and β be symmetric and monotone.

Then N_(λ,α,β) is an operator ideal.

Proof. Denote N :=N_(λ,α,β) for brevity.

1. Since e₁ ∈ λ,(I_K,0,0, . . .) ∈α(K^∗), e₁ ∈ β(K) and I_K = 1·I_K⊗1, we have I_K ∈ N.

2. Given Banach spaces X and Y, let us show that N(X, Y) is closed under addition. Let S1, S2 ∈ N(X, Y) and we pick representations

S1 =X

σ1x^∗1⊗y₁, S2 =X

σ2x^∗2⊗y₂. S₁+S₂ =X

(j₁σ₁+j₂σ₂)(j₁x^∗₁+j₂x^∗₂)⊗(j₁y₁+j₂y₂).

Since λ,α and β are symmetric and monotone, it follows from Lemma 1.28 that

j₁σ₁+j₂σ₂ ∈λ, j₁x^∗₁+j₂x^∗₂ ∈α, and j₁y₁+j₂y₂ ∈β.

In conclusion, S₁+S₂ ∈ N.

3. Let X, Y, V and W be Banach spaces. Consider operators S ∈ N(X, Y), R ∈ L(Y, V) and T ∈ L(W, X). We need to show that RST ∈ N(W, V). Pick a representation S = P

σx^∗ ⊗y with σ ∈ λ, x^∗ ∈α(X^∗), y∈β(Y). Then

RST =

∞

X

n=1

σnT^∗x^∗_n⊗Ryn.

Since (T^∗x^∗_n)∈α(W^∗) and (Ry_n)∈β(V), we have RST ∈ N.

Corollary 2.5. Let λ, µandν be symmetric and monotone sequence spaces.

Then both N_(λ,µ,ν) and N_(λ,µ^w_,ν^w₎ are operator ideals.

2.1 Nuclear quasi-norm

Let X be a Banach space and let q be a quasi-norm on a vector sequence space α(X).

Corollary 2.6. Let λ, µandν be symmetric and monotone sequence spaces.

Let µandν be equipped with K-symmetric and monotone quasi-norms. Then both N_(λ,ˆµw∗

,ˆν^w) and N_(λ,ˆµ^w_,ˆν^w₎ are operator ideals.

Definition 2.7. Let (λ, q_λ) be a quasi-normed sequence space, let (α, q_α) be a quasi-normed w^∗-sequence system and let (β, q_β) be a quasi-normed sequence system. Let X and Y be Banach spaces. Given an operator T ∈ N_(λ,α,β)(X, Y), we define thenuclear quasi-norm as

N(λ,α,β)(T) := infqλ(σ)qα(x^∗)qβ(y),

where the infimum ranges over all nuclear representations of T = Σσx^∗ ⊗y with σ ∈λ, x^∗ ∈α(X^∗) and y∈β(Y).

Theorem 2.8. Let (λ, q_λ) be a quasi-normed sequence space, let (α, q_α) be a quasi-normed w^∗-sequence system and let (β, q_β) be a quasi-normed sequence system. Let λ, α and β be symmetric and monotone, all equipped with nor-malized K-symmetric quasi-norms. Assume also that (λ, q_λ) ·(α(K), q_α)· (β(K), q_β) ≤ (`₁,k·k₁). Then (N_(λ,α,β), N_(λ,α,β)) is a quasi-normed operator ideal.

Proof. Denote N :=N_(λ,α,β) and N :=N_(λ,α,β) for brevity.

1. Let us show that N(I_K) = 1. Clearly,

N(I_K)≤q_λ(e₁)q_α((I_K,0,0, . . .))q_β(e₁) = 1, because the quasi-norms are normalized.

On the other hand, ifI_K =P

σx^∗⊗yfor some σ∈λ,x^∗ ∈α(K^∗) and y∈β(K), then

1 =I_K(1) =

∞

X

i=1

σ_ix^∗_i(1)y_i ≤ kσ·(j_K(1)◦x^∗)·yk₁ ≤

≤infq_λ(σ)q_α(x^∗)q_β(y) = N(I_K), In conclusion, N(I_K) = 1.

2. Let X and Y be Banach spaces and let κ_λ, κ_α, κ_β be the quasi-norm constants ofq_λ,q_α onα(X^∗)and q_β onβ(Y), respectively. Let S₁, S₂ ∈ N(X, Y). We shall show that N(S₁+S₂)≤4κ_λκ_ακ_β(N(S₁) +N(S₂)).

As in the proof of Theorem 2.4 pick representationsS1 =P

σ1x^∗1⊗y₁ and S2 =P

σ2x^∗2⊗y₂ and note that S :=S₁+S₂ =X

(j₁σ₁+j₂σ₂)(j₁x^∗₁ +j₂x^∗₂)⊗(j₁y₁+j₂y₂), where j₁ and j₂ are isometries due to K-symmetricity of the quasi-norms. We can assume that q_α(x^∗_k) = q_β(y_k) = 1 for k = 1,2. (If this does not hold for initial nuclear representations, we can normalize them as in Example 2.3.) Then

N(S)≤q_λ(j₁σ₁+j₂σ₂)·q_α(j₁x^∗₁+j₂x^∗₂)·q_β(j₁y₁+j₂y₂)≤

≤4κ_λκ_ακ_β(q_λ(σ₁) +q_λ(σ₂)).

This implies the claim.

3. Let X, Y, V and W be Banach spaces. Consider operators S ∈ N(X, Y), R ∈ L(Y, V) and T ∈ L(W, X). We need to show that N(RST) ≤ kRkN(S)kTk. This is immediate from the definition of a quasi-normed (w^∗-)sequence system and the representation RST = Pσ(T^∗◦x^∗)⊗(R◦y)(see the proof of Theorem 2.4) forS =P

σx^∗⊗y with σ ∈λ, x^∗ ∈α(X^∗),y ∈β(Y).

Corollary 2.9. Let λ, µ, ν be symmetric and monotone sequence spaces all equipped with K-symmetric normalized quasi-norms. Let µ and ν be FK-spaces or have monotone quasi-norms. Then both (N_(λ,ˆµw∗

,ˆν^w, N_(λ,ˆµw∗

)) and (N_(λ,ˆµ^w_,ˆν^w, N_(λ,ˆµ^w_,ˆν^w₎) are quasi-normed operator ideals.

Im Dokument λ-compact operators as a surjective hull of certain nuclear operators (Seite 16-21)