On Characterizations of the Space of p-Semi-Integral Multilinear Mappings

Note di Matematica 29, n. 1, 2009, 165–184.

On Characterizations of the Space of p-Semi-Integral Multilinear Mappings

Erhan C ¸ alı¸ skan

ⁱ

Department of Mathematics, Faculty of Sciences and Arts, Yıldız Technical University, 34210 Esenler, Istanbul, Turkey caliskan@yildiz.edu.tr

Received: 22/02/2008; accepted: 03/10/2008.

Abstract. In this paper we consider the ideal of p-semi-integral n-linear mappings, which is a natural multilinear extension of the ideal of p-summing linear operators. The space of p-semi-integral multilinear mappings is characterized by means of a suitable tensor norm up to an isometric isomorphism. In this connection we also consider tensor products of linear operators and multilinear mappings of finite type.

Keywords:p-semi integral multilinear mappings, tensor product of Banach spaces MSC 2000 classification:primary 46G25, secondary 46A32

Introduction

Semi-integral multilinear mappings between Banach spaces were introduced by R. Alencar and M. Matos [1] as a natural multilinear extension of the classical ideal of absolutely summing linear operators. The extension of this notion to p-semi-integral multilinear mappings, 1 ≤ p < +∞ is immediate [see [2, 11]].

It is shown in [11] that the class of p-semi-integral multilinear mappings has many good properties, e.g. the ideal property [11, Proposi¸c˜ao 5.1.11], inclusion property [11, Proposi¸c˜ao 5.1.9], etc. [see also [2]]. Also it follows from a result of V. Dimant [4] that p-semi integral multilinear mappings have good properties with respect to the Aron-Berner extensions. As well, R. Alencar and M. Matos in [1] show that every multilinear vector-valued Pietsch-integral mapping is semi integral. We refer to [2] and [11] for the relation between p-semi-integral multilinear mappings and other classes of p-summing multilinear mappings, such as dominated, multiple (or, fully), strongly and absolutely summing mappings.

The aim of this paper is to obtain characterizations of the space

L

si,p

(E

₁

, . . . , E

n

; F ) of p-semi-integral n-linear mappings from E

₁

× · · · × E

n

to F . In Section 2 we introduce a reasonable crossnorm σ

p

such that the space L

si,p

(E

₁

, . . . , E

n

; F

^

) of p-semi-integral n-linear mappings is isometric to the dual

iWe would like to thank Professor Daniel M. Pellegrino and Professor Geraldo Botelho for several helpful conversations and suggestions.

Note Mat. 29 (2009), n. 1, 165-184 ISSN 1123-2536, e-ISSN 1590-0932 DOI 10.1285/i15900932v29n1p165

http://siba-ese.unisalento.it, © 2009 Università del Salento

__________________________________________________________________

of E

₁

⊗ · · · ⊗ E

n

⊗ F endowed with σ

p

. A corresponding reasonable crossnorm σ

p

for scalar-valued p-semi-integral mappings is also studied. In Section 3 we study the continuity of the tensor product of linear operators with respect to the norm σ

p

(and σ

p

). Finally, in Section 4 we consider the norm σ

p

(and σ

p

) in connection with spaces of multilinear mappings of finite type. Stronger representation results are obtained for multilinear mappings of finite type on reflexive spaces.

The symbols E, E

₁

, . . . , E

n

, G

₁

, . . . , G

n,

F, F

₀

represent (real or complex) Ba- nach spaces, E

^

denotes the topological dual of E, K represents the scalar field and N represents the set of all positive integers. Given a natural number n ≥ 2, the Banach space of all continuous n-linear mappings from E

₁

× · · · × E

n

into F endowed with the sup norm will be denoted by L(E

1

, . . . , E

n

;F ) (L(E

1

, . . . , E

n

) if F = K). For p ≥ 1, l

p

(E) denotes the linear space of absolutely p-summable sequences (x

j

)

^∞_j=1

in E with the norm (x

j

)

^∞_j=1



p

= &

_∞

j=1

x

j



^p

)

¹_p

< ∞.

Also, l

_p^w

(E) denotes the linear space of the sequences (x

_j

)

^∞_j=1

in E such that (ϕ(x

_j

))

^∞_j=1

∈ l

p

for every ϕ ∈ E

^

. The expression

(x

j

)

^∞_j=1



w,p

= sup

ϕ∈BE´

(ϕ(x

j

))

^∞_j=1



p

defines a norm on l

^w_p

(E). If p = ∞ we are restricted to the case of bounded sequences and in l

∞

(E) we use the sup norm. The symbol E

1

⊗· · ·⊗E

n

denotes the algebraic tensor product of the Banach spaces E

1

, . . . , E

n

.

Let p ≥ 1. An n-linear mapping T ∈ L(E

1

, . . . , E

n

; F ) is p-semi-integral (T ∈ L

si,p

(E

₁

, . . . , E

n

; F )) if there exist C ≥ 0 and a regular probability measure μ on the Borel σ−algebra of B

_E^

1

× · · · × B

_E^_n

endowed with the product of the weak star topologies σ(E

^_l

, E

l

), l = 1, . . . , n, such that

T (x

1

, . . . , x

n

)  ≤ C

⎛

⎝ 

B_E1×···×B_En

|ϕ

1

(x

₁

) · · · ϕ

n

(x

n

) |

^p

dμ(ϕ

₁

, . . . , ϕ

n

)

⎞

⎠

1/p

for every x

j

∈ E

j

and j = 1, . . . , n. The infimum of the constants C working in the inequality defines a norm ·

_si,p

on L

si,p

(E

₁

, . . . , E

n

; F ).

1 p-Semi-Integral Mappings and Tensor Products of Banach Spaces

The following characterization of p-semi-integral mappings, which was proved

in [11] [see also [2]] will be important in this paper:

1 Theorem. [11], [2] Let E

₁

, . . . , E

n

and F be Banach spaces and let p ≥ 1.

Then, T ∈ L

si,p

(E

₁

, . . . , E

n

; F ) if and only if there exists C ≥ 0 such that

⎛

⎝ 

^m

j=1

T (x

1,j

, . . . , x

n,j

) 

^p

⎞

⎠

1/p

≤ C

⎛

⎜ ⎜

⎜ ⎝ sup

ϕ_l∈B l=1,...,nEl



m j=1

|ϕ

1

(x

_1,j

) · · · ϕ

n

(x

n,j

) |

^p

⎞

⎟ ⎟

⎟ ⎠

1/p

(1)

for every m ∈ N, x

l,j

∈ E

l

with l = 1, . . . , n and j = 1, . . . , m. Moreover, the infimum of the C in (1) is T 

si,p

.

A standard argument shows that L

si,p

(E

1

, . . . , E

n

; F ) is complete with re- spect to the norm  · 

si,p

. Next we introduce a reasonable crossnorm [see [14, p.

127]] on E

₁

⊗ · · · ⊗ E

n

⊗ F so that the topological dual of the resulting space is isometric to ( L

si,p

(E

₁

, . . . , E

_n

; F

^

),  · 

si,p

).

2 Proposition. Let E

₁

, . . . , E

n

and F be Banach spaces and let p ≥ 1. Let

σ

p

(u) := inf (λ

j

)

^m_j=1



q

⎛

⎜ ⎜

⎜ ⎝ sup

ϕ_l∈B l=1,...,nEl



m j=1

|ϕ

1

(x

_1,j

) · · · ϕ

n

(x

n,j

) |

^p

⎞

⎟ ⎟

⎟ ⎠

1/p

(b

j

)

^m_j=1



∞

where the infimum is taken over all representations of u ∈ E

1

⊗ · · · ⊗ E

n

⊗ F in the form

u =



m j=1

λ

j

x

_1,j

⊗ · · · ⊗ x

n,j

⊗ b

j

with m ∈ N, x

l,j

∈ E

l

, l = 1, . . . , n, λ

j

∈ K, b

j

∈ F , j = 1, . . . , m, and q ≥ 1 with 1/p + 1/q = 1.

Then the function σ

p

is a reasonable crossnorm on E

₁

⊗ · · · ⊗ E

n

⊗ F . For the proof we will need the following lemma.

3 Lemma. Given u ∈ E

1

⊗ · · · ⊗ E

n

⊗ F , for any δ > 0 we can find a representation of u of the form

u =



m j=1

α

j

x

_1,j

⊗ · · · ⊗ x

n,j

⊗ a

j

,

such that

 (α

j

)

^m_j=1



q

≤ [(1 + δ)σ

p

(u)]

^1/q

,

sup

ϕl∈B l=1,...,nEl



m j=1

| ϕ

1

(x

_1,j

) · · · ϕ

n

(x

n,j

) |

^p

≤ (1 + δ)σ

p

(u),

 (a

j

)

^m_j=1



∞

= 1.

Proof. Let us take a constant δ > 0. It is clear, by the definition of σ

p

, that we can choose a representation of u of the form

u =



m j=1

α

_j

x

_1,j

⊗ · · · ⊗ x

n,j

⊗ a

j

,

such that

σ

p

(u) ≤ (α

j

)

^m_j=1



q

⎛

⎜ ⎜

⎜ ⎝ sup

ϕ_l∈B l=1,...,nEl



m j=1

| ϕ

1

(x

_1,j

) · · · ϕ

n

(x

n,j

) |

^p

⎞

⎟ ⎟

⎟ ⎠

1/p

 (a

j

)

^m_j=1



∞

(*)

≤ (1 + δ)σ

p

(u) = [(1 + δ) σ

p

(u)]

^1/q

[(1 + δ) σ

p

(u)]

^1/p

.

Thus as a first step we can rearrange the representation of u by multiplying and dividing  (a

j

)

^m_j=1



∞

with a suitable constant c > 0 so that  (a

^∗j

)

^m_j=1



∞

:=  (ca

j

)

^m_j=1



∞

= 1, and  (α

^∗j

)

^m_j=1



q

:=  (

¹_c

α

j

)

^m_j=1



q

. Observe that the representa- tion u = 

^m

j=1

α

^∗_j

x

_1,j

⊗ · · · ⊗ x

n,j

⊗ a

^∗j

satisfies ( ∗) with

 (α

^∗j

)

^m_j=1



q

⎛

⎜ ⎜

⎜ ⎝ sup

ϕl∈B l=1,...,nEl



m j=1

| ϕ

1

(x

_1,j

) · · · ϕ

n

(x

n,j

) |

^p

⎞

⎟ ⎟

⎟ ⎠

1/p

≤ [(1 + δ)σ

p

(u)]

^1/q

[(1 + δ) σ

p

(u)]

^1/p

. Now as a second step, for this representation of u, for example, if

⎛

⎜ ⎜

⎜ ⎝ sup

ϕl∈B l=1,...,nEl



m j=1

| ϕ

1

(x

1,j

) · · · ϕ

n

(x

n,j

) |

^p

⎞

⎟ ⎟

⎟ ⎠

1/p

> [(1 + δ)σ

p

(u)]

^1/p

(**)

again we can choose a suitable constant C > 0 so that

⎛

⎜ ⎜

⎜ ⎝ sup

ϕl∈B l=1,...,nEl



m j=1

| ϕ

1

(Cx

_1,j

) · · · ϕ

n

(x

n,j

) |

^p

⎞

⎟ ⎟

⎟ ⎠

1/p

= [(1 + δ) σ

p

(u)]

^1/p

.

Hence, we have that

 (α

^∗j

)

^m_j=1



q

1 C

⎛

⎜ ⎜

⎜ ⎝ sup

ϕl∈B l=1,...,nEl



m j=1

| ϕ

1

(Cx

_1,j

) · · · ϕ

n

(x

_n,j

) |

^p

⎞

⎟ ⎟

⎟ ⎠

1/p

 (a

^∗j

)

^m_j=1



∞

≤ [(1 + δ)σ

p

(u)]

^1/q

[(1 + δ)σ

p

(u)]

^1/p

and this will imply that  (α

^∗j

)

^m_j=1



q 1

C

≤ [(1 + δ)σ

p

(u)]

^1/q

. Now taking

 (α

^∗∗j

)

^m_j=1



q

=  (

_C¹

α

^∗_j

)

^m_j=1



q

and x

^∗_1,j

= Cx

_1,j

, j = 1, . . . , m we obtain a representation of u of the form u = 

^m

j=1

α

^∗∗_j

x

^∗_1,j

⊗ · · · ⊗ x

n,j

⊗ a

^∗j

satisfying ( ∗) and conditions

 (α

^∗∗j

)

^m_j=1



q

≤ [(1 + δ)σ

p

(u)]

^1/q

,

sup

ϕl∈B l=1,...,nEl



m j=1

| ϕ

1

(x

^∗_1,j

) · · · ϕ

n

(x

_n,j

) |

^p

≤ (1 + δ)σ

p

(u),

 (a

^∗j

)

^m_j=1



∞

= 1.

Note that, in the second step above, if, instead of (**), it would be

 (α

^∗j

)

^m_j=1



q

> [(1 + δ) σ

p

(u)]

^1/q

, (***) then we would proceed completely in a similar way to obtain a suitable rep- resentation of u satisfying (*) and the above conditions. Note also that, as a consequence of the inequality (*), it cannot happen (**) and (***) simultane-

ously.

^QED

Proof of Proposition 2. First we show that σ

p

(u) = 0 implies u = 0.

Suppose that σ

p

(u) = 0. Then, for every  > 0, there is a representation



m j=1

λ

j

x

_1,j

⊗ · · · ⊗ x

n,j

⊗ b

j

of u such that

 (λ

j

)

^m_j=1



q

⎛

⎜ ⎜

⎜ ⎝ sup

ϕl∈B l=1,...,nEl



m j=1

| ϕ

1

(x

_1,j

) . . . ϕ

n

(x

n,j

) |

^p

⎞

⎟ ⎟

⎟ ⎠

1/p

 (b

j

)

^m_j=1



∞

< .

Hence it follows from the H¨older’s inequality that

sup

ϕl∈B_El,ϕ∈B_{F } l=1,...,n

%% %%

%% ϕ

₁

× · · · × ϕ

n

× ϕ(



m j=1

λ

_j

x

_1,j

⊗ · · · ⊗ x

n,j

⊗ b

j

)

%% %%

%%

= sup

ϕl∈B_El,ϕ∈B_{F } l=1,...,n

%% %%

%%



m j=1

ϕ

₁

(λ

_j

x

_1,j

) · · · ϕ

n

(x

_n,j

)ϕ(b

_j

)

%% %%

%%

≤ (b

j

)

^m_j=1

) 

∞

 (λ

j

)

^m_j=1

) 

q

⎛

⎜ ⎜

⎜ ⎝ sup

ϕl∈B l=1,...,nEl



m j=1

| ϕ

1

(x

_1,j

) · · · ϕ

n

(x

_n,j

) |

^p

⎞

⎟ ⎟

⎟ ⎠

1/p

< .

Thus we have that

%% %%

%%



m j=1

ϕ

₁

(λ

j

x

_1,j

) · · · ϕ

n

(x

n,j

)ϕ(b

j

)

%% %%

%% <   ϕ

1

 · · ·  ϕ

n

 ϕ , for every ϕ

l

∈ E

_l^

, l = 1, . . . , n and ϕ ∈ F

^

.

Since the value of the sum

%% %%

%% ϕ

₁

× · · · × ϕ

n

× ϕ(



m j=1

λ

j

x

_1,j

⊗ · · · ⊗ x

n,j

⊗ b

j

)

%% %%

%%

is independent of the representation of u, it follows that



m j=1

ϕ

₁

(λ

j

x

_1,j

) · · · ϕ

n

(x

n,j

)ϕ(b

j

) = 0,

for every ϕ

l

∈ E

_l^

, l = 1, . . . , n, ϕ ∈ F

^

.

Hence, since E

₁^

, . . . , E

_n^

and F

^

are separating subsets of the respective al- gebraic duals, by the multilinear version of [14, Proposition 1.2] it follows that u = 0.

To prove the triangular inequality, take u, v ∈ E

1

⊗ · · · ⊗ E

n

⊗ F . For any δ > 0, by Lemma 3 we can find representations

u =



m j=1

α

_j

x

_1,j

⊗ · · · ⊗ x

n,j

⊗ a

j

and v =



m j=1

β

_j

y

_1,j

⊗ · · · ⊗ y

n,j

⊗ b

j

such that

(α

j

)

^m_j=1



q

≤ [(1 + δ)σ

p

(u)]

^1/q

,

(β

j

)

^m_j=1



q

≤ [(1 + δ)σ

p

(v)]

^1/q

,

sup

ϕl∈B l=1,...,nEl



m j=1

|ϕ

1

(x

_1,j

) · · · ϕ

n

(x

_n,j

) |

^p

≤ (1 + δ)σ

p

(u),

sup

ϕ_l∈B l=1,...,nEl



m j=1

|ϕ

1

(y

_1,j

) · · · ϕ

n

(y

n,j

) |

^p

≤ (1 + δ)σ

p

(v),

(a

j

)

^m_j=1



∞

= 1 = (b

j

)

^m_j=1



∞

. Then it follows that

σ

p

(u + v) ≤

⎛

⎝ 

^m

j=1

|α

j

|

^q

+



m j=1

|β

j

|

^q

⎞

⎠

1/q

×

⎛

⎜ ⎜

⎜ ⎝ sup

ϕ_l∈B l=1,...,nEl

⎛

⎝ 

^m

j=1

|ϕ

1

(x

1,j

) · · · ϕ

n

(x

n,j

)|

^p

+



m j=1

|ϕ

1

(y

1,j

) · · · ϕ

n

(y

n,j

)|

^p

⎞

⎠

⎞

⎟ ⎟

⎟ ⎠

1/p

≤ (1 + δ)

^1/q

(σ

p

(u) + σ

p

(v))

^1/q

(1 + δ)

^1/p

(σ

p

(u) + σ

p

(v))

^1/p

= (1 + δ)( σ

p

(u) + σ

p

(v)),

which shows the triangular inequality. Hence σ

p

is a norm on E

₁

⊗ · · · ⊗ E

n

⊗ F . It is easily seen that σ

p

(x

₁

⊗· · ·⊗x

n

⊗b) ≤ x

1

 · · · x

n

·b for every x

l

∈ E

l

, l = 1, . . . , n and b ∈ F . To show that ϕ

1

⊗ · · · ⊗ ϕ

n

⊗ ϕ ≤ ϕ

1

 · · · ϕ

n

 · ϕ

let ϕ

l

∈ E

l^

with ϕ

l

= 0, l = 1, . . . , n, let ϕ ∈ F

^

with ϕ = 0, and let u =



m j=1

λ

_j

x

_1,j

⊗ · · · ⊗ x

n,j

⊗ b

j

. Then by the H¨older’s inequality we get

|ϕ

1

⊗ · · · ⊗ ϕ

n

(u) | ≤ ϕ(b

j

)

^m_j=1



∞

ϕ

1

 · · · ϕ

n

(λ

j

)

^m_j=1



q

×

⎛

⎜ ⎜

⎜ ⎝ sup

ϕl∈B l=1,...,nEl



m j=1

|ϕ

1

(x

_1,j

) · · · ϕ

n

(x

_n,j

) |

^p

⎞

⎟ ⎟

⎟ ⎠

1/p

.

Therefore we obtain that |ϕ

1

⊗ · · · ⊗ ϕ

n

⊗ ϕ(u)| ≤ ϕ

1

 · · · ϕ

n

ϕσ

p

(u), and

we have shown that σ

p

is a reasonable crossnorm.

^QED

Note that when n = 1, in particular, the norm σ

p

is reduced to the Chevet- Saphar norm d

q

on E

₁

⊗ F [see [14, pg. 135]].

In the previous proposition if we take F = K, then we identify E

1

⊗· · ·⊗E

n

⊗ K with E

1

⊗ · · · ⊗ E

n

, and in this case the corresponding reasonable crossnorm will be denoted by σ

p

which is described as follows:

σ

_p

(u) := inf (λ

j

)

^m_j=1



q

⎛

⎜ ⎜

⎜ ⎝ sup

ϕl∈B l=1,...,nEl



m j=1

|ϕ

1

(x

_1,j

) · · · ϕ

n

(x

_n,j

) |

^p

⎞

⎟ ⎟

⎟ ⎠

1/p

where the infimum is taken over all representations of u ∈ E

1

⊗ · · · ⊗ E

n

in the form u = 

^m

j=1

λ

j

x

_1,j

⊗ · · · ⊗ x

n,j

with m ∈ N, x

l,j

∈ E

l

, l = 1, . . . , n, λ

j

∈ K, j = 1, . . . , m, and q ≥ 1 with

¹_p

+

¹_q

= 1.

4 Remark. (Commutativity and associativity of σ

_p

) Let E, F and G be Banach spaces. Since the algebraic isomorphisms E ⊗ F = F ⊗ E and E ⊗ (F ⊗ G) = (E ⊗ F ) ⊗ G are well known [see, for example, [7, p. 179]] then it follows by the very definition of σ

_p

that, the normed (resp. Banach) spaces (E ⊗ F, σ

p

) and (F ⊗ E, σ

p

) (resp. (E ˜ ⊗F, σ

p

) and (F ˜ ⊗E, σ

p

)) are isometrically isomorphic, and the normed (resp. Banach) spaces ((E ⊗ F, σ

p

) ⊗ G, σ

p

) and (E ⊗ (F ⊗ G, σ

p

), σ

_p

) (resp. ((E ˜ ⊗F, σ

p

) ˜ ⊗G, σ

p

) and (E ˜ ⊗(F ˜⊗G, σ

p

), σ

_p

)) are isometrically isomorphic in the canonical way, where the symbol ˜ ⊗ denotes the completion of the corresponding normed space.

The above remark assures that the (reasonable) crossnorm σ

p

is symmetric, that is, if we interchange the factor spaces the value of the norm does not alter.

Although σ

p

and σ

p

share many properties, let us see that, contrary to the case of σ

p

, commutativity and associativity do not hold for σ

p

: take a tensor u in E ⊗ F and consider the infima

inf (λ

j

)

^m_j=1



q

⎛

⎝ sup

ϕ∈B_E



m j=1

|ϕ(x

j

) |

^p

⎞

⎠

1/p

(y

j

)

^m_j=1



∞

and

inf (λ

j

)

^m_j=1



q

⎛

⎝ sup

φ∈B_{F }



m j=1

|φ(y

j

) |

^p

⎞

⎠

1/p

(x

j

)

^m_j=1



∞

,

where the infima are taken over all representations u = 

^m

j=1

λ

j

x

j

⊗ y

j

with

λ

j

∈ K, x

j

∈ E, y

j

∈ F , j = 1, . . . , m. The fact that these infima are different

in general shows that σ

p

is not a symmetric norm. Its non-associativity follows analogously.

5 Remark. Let E

₁

, . . . , E

_n

and F be Banach spaces and let p ≥ 1.

(a) It follows from the definitions of σ

_p

and σ

p

that σ

_p

(u) ≤ σ

p

(u) for every u ∈ E

1

⊗ · · · ⊗ E

n

⊗ F .

(b) To each tensor u ∈ E

₁^

⊗· · ·⊗E

n^

corresponds a canonical operator T

_u

: E

₁

×

· · · × E

n

−→ K given by

u =



m j=1

λ

j

ϕ

_1,j

⊗ · · · ⊗ ϕ

n,j

→ T

u

=



m j=1

λ

j

ϕ

_1,j

× · · · × ϕ

n,j

,

with λ

_j

∈ K, ϕ

l,j

∈ E

_l^

, l = 1, . . . , n, j = 1, . . . , m. By an easy application of H¨older’s inequality we see that T

u

 ≤ σ

p

(u) for every u ∈ E

₁^

⊗· · ·⊗E

^n

. Below by combining the argument of the proof of [9, Theorem 3.7] with Theorem 1 we prove the following result. This result characterizes the space of p-semi integral mappings as the topological dual of the space of the tensor product (E

₁

⊗ · · · ⊗ E

n

⊗ F, σ

p

) up to an isometric isomorphism.

6 Proposition. Let E

₁

, . . . , E

n

be Banach spaces. Then, for every Banach space F and p ≥ 1, the space (L

si,p

(E

₁

, . . . , E

n

; F

^

),  · 

si,p

) is isometrically isomorphic to (E

₁

⊗ · · · ⊗ E

n

⊗ F, σ

p

)

^

through the mapping T −→ φ

T

, where φ

T

(x

₁

⊗ · · · ⊗ x

n

⊗ b) = T (x

1

, . . . , x

n

)(b), for every x

l

∈ E

l

, l = 1, . . . , n, and b ∈ F .

Proof. It is easy to see that the correspondence

T ∈ L

si,p

(E

₁

, . . . , E

n

; F

^

) −→ φ

T

∈ (E

1

⊗ · · · ⊗ E

n

⊗ F, σ

p

)

^

defined by

φ

_T

(x

₁

⊗ · · · ⊗ x

n

⊗ b) := T (x

1

, . . . , x

_n

)(b), x

_l

∈ E

l

, l = 1, . . . , n and b ∈ F, is linear and injective. To show the surjectivity let φ ∈ (E

1

⊗· · ·⊗E

n

⊗F, σ

p

)

^

and consider the corresponding n-linear mapping T

_φ

∈ L(E

1

, . . . , E

n

; F

^

), defined by T

_φ

(x

₁

, . . . , x

n

)(b) = φ(x

₁

⊗ · · · ⊗ x

n

⊗ b), for x

l

∈ E

l

, l = 1, . . . , n, and b ∈ F . Let us consider x

_l,j

∈ E

l

, l = 1, . . . , n, j = 1, . . . , m. For every  > 0 there are b

j

∈ F , with b

j

 = 1, j = 1, . . . , m, such that

(T

φ

(x

1,j

, . . . , x

n,j

))

^m_j=1



^pp

=



m j=1

T

φ

(x

1,j

, . . . , x

n,j

)

^p

≤  +



m j=1

|T

φ

(x

_1,j

, . . . , x

_n,j

)(b

_j

) |

^p

= ( ∗).

Now we can choose λ

j

∈ K, with |λ

j

| = 1, j = 1, . . . , m, such that

( ∗) =  +



m j=1

|φ(x

1,j

⊗ · · · ⊗ x

n,j

⊗ b

j

) |

^p

=  +

%% %%

%%



m j=1

|φ(x

1,j

⊗ · · · ⊗ x

n,j

⊗ b

j

)|

^p−1

λ

j

φ(x

_1,j

⊗ · · · ⊗ x

n,j

⊗ b

j

)

%% %%

%% = (∗∗).

Proceeding from this point, by continuity of φ and the H¨older’s inequality we get

(∗∗) ≤ + φ

(E1⊗···⊗En⊗F,σp)^

σ

p

⎛

⎝ 

^m

j=1

λ

j

|φ(x

1,j

⊗ · · · ⊗ x

n,j

⊗ b

j

)|

^p−1

x

_1,j

⊗ · · · ⊗ x

n,j

⊗ b

j

⎞

⎠

≤ + φ

(E1⊗···⊗En⊗F,σp)^

  0

λ

j

|φ(x

1,j

⊗ · · · ⊗ x

n,j

⊗ b

j

) |

^p−1

1

m j=1

 

q

×

⎛

⎜ ⎜

⎜ ⎝ sup

ϕ_l∈B l=1,...,nEl



m j=1

|ϕ

1

(x

_1,j

) · · · ϕ

n

(x

n,j

) |

^p

⎞

⎟ ⎟

⎟ ⎠

1/p

(b

j

)

^m_j=1



∞

= + φ

(E1⊗···⊗En⊗F,σp)^

 (T

φ

(x

_1,j

, . . . , x

n,j

))

^m_j=1

 

^p/q

⎛

p

⎜ ⎜

⎜ ⎝ sup

ϕl∈B l=1,...,nEl



m j=1

|ϕ

1

(x

1,j

) · · · ϕ

n

(x

n,j

)|

^p

⎞

⎟ ⎟

⎟ ⎠

1/p

.

Since  is arbitrary and p − (p/q) = 1 we obtain

(T

φ

(x

_1,j

, . . . , x

_1,j

))

^m_j=1



p

≤ φ

(E1⊗···⊗En⊗F,σp)^

⎛

⎜ ⎜

⎜ ⎝ sup

ϕ_l∈B l=1,...,nEl



m j=1

|ϕ

1

(x

_1,j

) · · · ϕ

n

(x

n,j

) |

^p

⎞

⎟ ⎟

⎟ ⎠

1/p

,

showing that T

φ



si,p

≤ φ

(E1⊗···⊗En⊗F,σp)^

, and therefore

T

φ

∈ (L

si,p

(E

₁

, . . . , E

n

; F

^

),  · 

si,p

).

To show the reverse inequality let T ∈ L

si,p

(E

₁

, . . . , E

n

; F

^

) and consider the linear functional φ

T

on E

₁

⊗ · · · ⊗ E

n

⊗ F given by

φ

_T

(u) =



m j=1

λ

_j

T (x

_1,j

, . . . , x

_n,j

)(b

_j

)

for u =



m j=1

λ

j

x

_1,j

⊗ · · · ⊗ x

n,j

⊗ b

j

, where m ∈ N, λ

j

∈ K, k = 1, . . . , n, b

j

∈ F , j = 1, . . . , m. Hence, by H¨older’s inequality and Theorem 1 it follows that

|φ

T

(u) |

^p

≤ (λ

j

)

^m_j=1



^pq

(b

j

)

^m_j=1



^p∞

T 

^p_si,p

sup

ϕl∈B l=1,...,nEl



m j=1

|ϕ

1

(x

_1,j

) · · · ϕ

n

(x

_n,j

) |

^p

.

Thus |φ

T

(u)| ≤ T 

si,p

σ

p

(u), showing that φ

T

is σ

p

-continuous with

φ

T



(E1⊗···⊗En⊗F,σp)^

≤ T 

si,p

.

^QED

Making F = K, in the previous Proposition we obtain that for every Ba- nach spaces E

₁

, . . . , E

n

, and p ≥ 1, the space of p-semi-integral forms (L

si,p

(E

₁

, . . . , E

n

), · 

si,p

) is isometric to (E

₁

⊗ · · · ⊗ E

n

⊗ K, σ

p

)

^

.

On the other hand, by a slight modification of the proof of Proposition 6, alternatively, we obtain the representation of the space of p-semi-integral forms as the dual of the tensor product endowed with the σ

p

-norm.

7 Proposition. Let E

₁

, . . . , E

_n

be Banach spaces, and let p ≥ 1. Then ( L

si,p

(E

₁

, . . . , E

n

),  · 

si,p

) is isometrically isomorphic to (E

₁

⊗ · · · ⊗ E

n

, σ

p

)

^

through the mapping T −→ φ

T

, where φ

_T

(x

₁

⊗ · · · ⊗ x

n

) = T (x

₁

, . . . , x

n

) for every x

_l

∈ E

l

, l = 1, . . . , n.

It is interesting to observe that (E

1

⊗ · · · ⊗ E

n

⊗ K, σ

p

)

^

is not isometric to (E

1

⊗ · · · ⊗ E

n

, σ

p

)

^

, but as a consequence of Propositions 6 and 7 we see that (E

1

⊗ · · · ⊗ E

n

⊗ K, σ

p

)

^

is isometric to (E

1

⊗ · · · ⊗ E

n

, σ

p

)

^

.

Recall that a linear operator u : E −→ F is said to be absolutely p-summing if (u(x

_j

))

^∞_j=1

∈ l

p

(F ) whenever (x

_j

)

^∞_j=1

∈ l

^wp

(E). The vector space (operator ideal) composed by all absolutely p-summing operators from E to F is denoted by L

as,p

(E; F ). Hence the class of absolutely p-summing linear mappings coin- cides with the class of p-semi integral linear mappings. So in the linear case we prefer to write L

as,p

(E; F ) (resp.  . 

as,p

) instead of L

si,p

(E; F ) (resp.  . 

si,p

).

For the theory of absolutely summing operators we refer to [3].

Below, inspired by a result of D. P´erez-Garc´ıa [12], we show that the norm σ

_p

is well behaved in connection with p-semi integral mappings.

8 Proposition. Let E

₁

, . . . , E

n

and F be Banach spaces and let p ≥ 1.

Then we have the following:

(a) If T : E

₁

⊗ · · · ⊗ E

n

−→ F is a linear operator, then

T ∈ L((E

1

⊗ · · ·   ⊗E

n

, σ

p

); F ) if and only if ϕ ◦ T ∈ (E

1

⊗ · · ·   ⊗E

n

, σ

p

)

^

for every ϕ ∈ B

F^

. In this case we have:

 T 

_{L((E1⊗···⊗ En,σp);F )}

= sup

ϕ∈B_{F }

 ϕ ◦ T 

_{(E1⊗···⊗ En,σp)}

.

(b) A multilinear mapping T : E

₁

× · · · × E

n

−→ F is p-semi integral if its associated linear mapping  T : E

₁

⊗ · · · ⊗ E

n

−→ F , given by  T (x

₁

⊗ · · · ⊗ x

n

) = T (x

₁

, . . . , x

n

) for every x

l

∈ E

l

, l = 1, . . . , n, is σ

p

-continuous and p-semi integral. In this case we have

 T ≤ T 

si,p

≤  T 

si,p

.

Conversely, if T ∈ L

si,p

(E

₁

, . . . , E

_n

; F ), then the associated linear mapping T is σ 

p

-continuous, that is,  T ∈ L((E

1

⊗ · · · ⊗ E

n

, σ

p

); F ). In this case we have:

 T ≤  T 

L((E1⊗···⊗En,σp);F )

≤ T 

si,p

.

Proof. (a) The non-trivial implication of the first assertion is an easy consequence of the closed graph theorem. To show the second assertion let u

₀

∈ E

1

⊗ · · · ⊗ E

n

with T u

₀

= 0. Then by the Hanh-Banach Theorem there exists a ϕ

₀

∈ B

F^

such that ϕ

₀

(T u

₀

) =  T u

0

. Therefore for every ϕ ∈ B

F^

we have that

 T u

0

≤ sup

ϕ∈B_{F }

| ϕ ◦ T (u

0

) |≤ sup

ϕ∈B_{F }

 ϕ ◦ T 

_{(E1⊗···⊗ En,σp)}

σ

p

(u

₀

),

which shows that

 T 

_{L((E1⊗···⊗ En,σp);F )}

≤ sup

ϕ∈B_{F }

 ϕ ◦ T 

_{(E1⊗···⊗ En,σp)}

.

Since the reverse inequality is immediate we have (a).

(b) Suppose  T ∈ L

si,p

((E

₁

⊗ · · · ⊗ E

n

, σ

p

); F ). Then by Proposition 7 and

Theorem 1 it follows that

⎛

⎝ 

^m

j=1

T (x

1,j

, . . . , x

n,j

) 

^p

⎞

⎠

1/p

≤  T 

si,p

⎛

⎝ sup

ϕ∈B(E1⊗···⊗En,σp)



m j=1

|ϕ(x

1,j

⊗ · · · ⊗ x

n,j

) |

^p

⎞

⎠

1/p

=  T 

si,p

⎛

⎝ sup

S∈B(Lsi,p(E1,··· ,En),.si,p)



m j=1

|S(x

1,j

, · · · , x

n,j

)|

^p

⎞

⎠

1/p

≤  T 

si,p

⎛

⎜ ⎜

⎜ ⎝ sup

ϕ_l∈B l=1,...,nEl



m j=1

|ϕ

1

(x

_1,j

) · · · ϕ

n

(x

n,j

)|

^p

⎞

⎟ ⎟

⎟ ⎠

1/p

,

which shows that T ∈ L

si,p

(E

₁

, . . . , E

n

; F ) with  T 

si,p

≤  T 

si,p

. The fact that  T ≤ T 

si,p

follows easily from Theorem 1.

To show the converse, suppose now T ∈ L

si,p

(E

₁

, . . . , E

n

; F ), and let u ∈ E

₁

⊗ · · · ⊗ E

n

. Choosing a representation u = 

m

j=1

λ

j

x

_1,j

⊗ · · · ⊗ x

n,j

, from the H¨older’s inequality and Theorem 1 it follows that

  T (u) 

^p

≤ (λ

j

)

^m_j=1



^pq



m j=1

 T (x

1,j

, . . . , x

_n,j

) 

^p

≤ (λ

j

)

^m_j=1



^pq

T 

^psi,p

sup

ϕl∈B l=1,...,nEl



m j=1

|ϕ

1

(x

_1,j

) · · · ϕ

n

(x

n,j

) |

^p

.

Hence   T (u)  ≤ T 

si,p

σ

_p

(u), and so  T is σ

_p

-continuous with

  T 

L((E1⊗···⊗En,σp);F )

≤ T 

si,p

. Finally, since σ

p

is a reasonable cross- norm, it readily follows that  T ≤  T 

L((E1⊗···⊗En,σp);F )

, which com- pletes the proof of (b).

QED

Proposition 8(b) can be seen as a weak vector-valued version of Proposition 7. We do not know if, in general,  T ∈ L

si,p

((E

₁

⊗ · · · ⊗ E

n

, σ

p

); F ) whenever T ∈ L

si,p

(E

₁

, . . . , E

n

; F ).

We end this section by giving another property of the p-semi integral mul-

tilinear mappings.

9 Proposition. [11, Teorema 5.1.14] If T ∈ L

si,p

(E

₁

, . . . , E

n

; F ) then, for each i = 1, . . . , n, the mapping T

_i

: E

_i

−→ L(E

1

, . . ., E

^[i] _n

; F ), defined by T

i

(x

i

)(x

₁

, . . ., x

^[i] n

) := T (x

₁

, . . . , x

n

), is absolutely p-summing with

T

i

(x

i

) ∈ L

si,p

(E

₁

, . . ., E

^[i] n

; F ). Furthermore,

 T = T

i

≤ T

i



as,p

≤ T 

si,p

.

Proof. A close examination of the proof of [11, Teorema 5.1.14] gives the first part. Since it is readily seen that  T = T

i

 and, it follows by Proposition 8(b) that  T

i

≤ T

i



si,p

, we have the proof.

^QED

2 Tensor Product of Operators

In this section we consider the tensor product of linear operators in connec- tion with the reasonable crossnorm σ

p

(and σ

p

). We show that the reasonable crossnorms σ

p

and σ

p

are actually tensor norms. The results of this section are similar to those ones given for the projective tensor product in connection with bilinear mappings in [14] with the same patterns in corresponding proofs [see [14, Propositions 2.3 and 2.4]].

In what follows we use the notation σ

p;E1,...,En

to emphasize that the cross- norm σ

p

is considered on E

₁

⊗ · · · ⊗ E

n

.

10 Proposition. Let T

i

∈ L(E

i

; F

i

), i = 1, . . . , n, T ∈ L(E; F ) and p ≥ 1. Then there is a unique continuous linear operator T

₁

⊗

σp

· · · ⊗

σp

T

n

⊗

σp

T : (E

₁

⊗ · · · ˜⊗E ˜

n

⊗E, σ ˜

p

) −→ (F

1

⊗ · · · ˜⊗F ˜

n

⊗F, σ ˜

p

) such that

T

₁

⊗

σp

· · · ⊗

σp

T

_n

⊗

σp

T (x

₁

⊗ · · · ⊗ x

n

⊗ x) = (T

1

x

₁

) ⊗ · · · ⊗ (T

n

x

_n

) ⊗ (T x) for every x

i

∈ E

i

, i = 1, . . . , n, and x ∈ E. Moreover

T

1

⊗

σp

· · · ⊗

σp

T

n

⊗

σp

T  = T

1

⊗ · · · ⊗ T

n

⊗ T  = T

1

 · · · T

n

T .

Proof. Given linear operators T

_i

∈ L(E

i

; F

_i

), i = 1, . . . , n, and T ∈ L(E; F ), there is a unique linear operator T

1

⊗· · ·⊗T

n

⊗T : E

1

⊗· · ·⊗E

n

⊗E −→

F

₁

⊗ · · · ⊗ F

n

⊗ F such that

T

1

⊗ · · · ⊗ T

n

⊗ T (x

1

⊗ · · · ⊗ x

n

⊗ x) = (T

1

x

1

) ⊗ · · · ⊗ (T

n

x

n

) ⊗ (T x) for every x

i

∈ E

i

, i = 1, . . . , n and x ∈ E [see [14, p. 7]]. We may suppose T

i

= 0, i = 1, . . . , n and T = 0. Let u ∈ E

1

⊗· · ·⊗E

n

⊗E and let



m j=1

λ

j

x

_1,j

⊗· · ·⊗x

n,j

⊗x

j

be a representation of u. Hence the sum



m j=1

λ

j

T

₁

(x

_1,j

) ⊗ · · · ⊗ T

n

(x

n,j

) ⊗ T (x

j

)

is a representation of T

₁

⊗ · · · ⊗ T

n

⊗ T (u) in F

1

⊗ · · · ⊗ F

n

⊗ F . Then, for every p ≥ 1

σ

p;F1,...,Fn,F

(T

₁

⊗ · · · ⊗ T

n

⊗ T (u))

≤ (λ

j

)

^m_j=1



q

⎛

⎜ ⎜

⎜ ⎝ sup

φl∈B_{F l} l=1,...,n



m j=1

|φ

1

(T

₁

x

_1,j

) . . . φ

_n

(T

_n

x

_n,j

) |

^p

⎞

⎟ ⎟

⎟ ⎠

1/p

(T x

j

)

^m_j=1



∞

≤ T

1

 · · · T

n

T (λ

j

)

^m_j=1



q

⎛

⎜ ⎜

⎜ ⎝ sup

φ_l∈B_El l=1,...,n



m j=1

|φ

1

(x

_1,j

) . . . φ

n

(x

n,j

)|

^p

⎞

⎟ ⎟

⎟ ⎠

1/p

(x

j

)

^m_j=1



∞

and we have that

σ

p;F1,...,Fn,F

(T

₁

⊗ · · · ⊗ T

n

⊗ T (u)) ≤ T

1

 · · · T

n

T σ

p;E1,...,En,E

(u), so that the linear operator T

₁

⊗ · · · ⊗ T

n

⊗ T is continuous for the crossnorms on E

₁

⊗· · ·⊗E

n

⊗E and F

1

⊗· · ·⊗F

n

⊗F and T

1

⊗· · ·⊗T

n

⊗T  ≤ T

1

 · · · T

n

T .

On the other hand, as σ

p

is an reasonable crossnorm we get that

T

1

(x

₁

) · · · T

n

(x

n

)T (x) = σ

p;F1,...,Fn,F

(T

₁

(x

₁

) ⊗ · · · ⊗ T

n

(x

n

) ⊗ T (x))

≤ T

1

⊗ · · · ⊗ T

n

⊗ T  σ

p;E1,...,En,E

(x

₁

⊗ · · · ⊗ x

n

⊗ x)

= T

1

⊗ · · · ⊗ T

n

⊗ T x

1

 · · · x

n

x, [see [14, Proposition 6.1]], and therefore T

1

⊗· · ·⊗T

n

⊗T  ≥ T

1

 · · · T

n

T .

Hence we have that

T

1

⊗ · · · ⊗ T

n

⊗ T  = T

1

 · · · T

n

T 

Now taking the unique continuous extension of the operator T

₁

⊗ · · · ⊗ T

n

⊗ T to the completions of (E

₁

⊗ · · · ⊗ E

n

⊗ E, σ

p

) and (F

₁

⊗ · · · ⊗ F

n

⊗ F, σ

p

), which we denote by T

₁

⊗

σp

· · · ⊗

σp

T

_n

⊗

σp

T , we obtain a unique linear operator from (E

₁

⊗ · · · ˜⊗E ˜

n

⊗E, σ ˜

p

) into (F

₁

⊗ · · · ˜⊗F ˜

n

⊗F, σ ˜

p

) with the norm T

1

⊗

σp

· · · ⊗

σp

T

_n

⊗

σp

T  = T

1

 · · · T

n

T .

^QED

The σ

p

-tensor product does not respect subspaces but respects 1-comple-

mented subspaces. Indeed; if E

₀

is a subspace of E, then E

₀

⊗ F is an algebraic

subspace of E ⊗ F , but the norm induced on E

0

⊗ F by (E ⊗ F, σ

p

) is not, in

general the σ

p

norm on E

₀

⊗ F . In fact, if we take u ∈ E

0

⊗ F , then we see that

σ

p;E,F

(u) = inf (λ

j

)

^m_j=1



q

⎛

⎝ sup

ϕ∈B_E



m j=1

|ϕ(x

j

) |

^p

⎞

⎠

1/p

(y

j

)

^m_j=1



∞

≤ inf (λ

j

)

^m_j=1



q

⎛

⎝ sup

ψ∈B_E0



m j=1

|ψ(x

j

)|

^p

⎞

⎠

1/p

(y

j

)

^m_j=1



∞

= σ

p;E0,F

(u)

since the set of representations of u become bigger when we enlarge the space E

₀

to E. Similarly if F

₀

is a subspace of F , then E ⊗F

0

is an algebraic subspace of E ⊗ F , but the norm induced on E ⊗ F

0

by (E ⊗ F, σ

p

) is not, in general the

σ

p

norm on E ⊗ F

0

. Whereas for complemented subspaces we have:

11 Proposition. Let M

₁

, . . . , M

n

, N be complemented subspaces of E

₁

, . . . , E

n

, F respectively. Then M

₁

⊗ · · · ⊗ M

n

⊗ N is complemented in (E

1

⊗

· · · ⊗ E

n

⊗ F, σ

p

) and the norm on M

₁

⊗ · · · ⊗ M

n

⊗ N induced by σ

p;E1,...,En,F

is equivalent to σ

p;M1,...,Mn,N

. Moreover, if M

₁

, . . . , M

n

and N are 1-complemented, then (M

₁

⊗ · · · ⊗ M

n

⊗ N, σ

p

) is 1-complemented in (E

₁

⊗ · · · ⊗ E

n

⊗ F, σ

p

) as well.

Proof. Let P

₁

, . . . , P

n

, Q be projections from E

₁

, . . . , E

n

, F onto M

₁

, . . . , M

n

, N respectively. One can easily show that P

₁

⊗ · · · ⊗ P

n

⊗ Q is a projection of (E

₁

⊗· · ·⊗E

n

⊗F, σ

p

) onto M

₁

⊗· · ·⊗M

n

⊗N. We just have proved above that σ

p;E,F

(u) ≤ σ

p;M,N

(u) for u ∈ M ⊗N, and the same argument shows that σ

p;E1,...,En,F

(u) ≤ σ

p;M1,...,Mn,N

(u) for u ∈ M

1

⊗ · · · ⊗ M

n

⊗ N.

Let u ∈ M

1

⊗· · ·⊗M

n

⊗N and let 

m

j=1

λ

j

x

_1,j

· · ·⊗x

n,j

⊗y

j

be a representa- tion of u in E

₁

⊗· · ·⊗E

n

⊗F . Then u = P

1

⊗· · ·⊗P

n

⊗Q(u) = 

m

j=1

λ

j

P

₁

(x

_1,j

)⊗

· · · ⊗ P

n

(x

n,j

) ⊗ Q(y

j

) is a representation of u in M

₁

⊗ · · · ⊗ M

n

⊗ N. Therefore, by the argument used in the proof of Proposition 10 we obtain

σ

p;M1,...,Mn,N

(u)

≤ (λ

j

)

^m_j=1



q

⎛

⎜ ⎜

⎜ ⎝ sup

φ_l∈B_Ml l=1,...,n



m j=1

|φ

1

(P

₁

(x

_1,j

)) · · · φ

n

(P

n

(x

n,j

))|

^p

⎞

⎟ ⎟

⎟ ⎠

1/p

(Q(y

j

))

^m_j=1



∞

≤ P

1

 · · · P

n

Q(λ

j

)

^m_j=1



q

⎛

⎜ ⎜

⎜ ⎝ sup

φl∈B_El l=1,...,n



m j=1

|φ

1

(x

_1,j

) · · · φ

n

(x

_n,j

) |

^p

⎞

⎟ ⎟

⎟ ⎠

1/p

(y

j

)

^m_j=1



∞

.

Since this holds for every representation of u in E

₁

⊗ · · · ⊗ E

n

⊗ F , it follows that

σ

p;E1,...,En,F

(u) ≤ σ

p;M1,...,Mn,N

(u) ≤ P

1

 · · · P

n

Qσ

p;E1,...,En,F

(u).

Now, if M

₁

, . . . , M

_n

and N are complemented by projections of norm one, then we have that σ

p;E1,...,En,F

(u) = σ

p;M1,...,Mn,N

(u) for every u ∈ M

1

⊗· · ·⊗M

n

⊗N, and by Proposition 10 it follows that P

1

⊗· · ·⊗P

n

⊗Q = P

1

 · · · P

1

Q = 1,

as we desired.

^QED

We note that an analogous result to Proposition 10, in a similar way, can be obtained for σ

p

also. As well, like the case of σ

p

, and with analogous reasonings, the σ

p

-tensor product does not respect subspaces but respects 1-complemented subspaces.

3 Connection with multilinear mappings of finite type

We recall that a multilinear mapping T ∈ L(E

1

, . . . , E

_n

; F ) is said to be of finite type if it has a finite representation of the form

T =



m j=1

λ

j

ϕ

1,j

× · · · × ϕ

n,j

b

j

(2)

where λ

_j

∈ K, ϕ

l,j

∈ E

_l^

, l = 1, . . . , n, b

_j

∈ F , j = 1, . . . , m. We denote by L

f

(E

₁

, . . . , E

_n

; F ) the vector subspace of L(E

1

, . . . , E

_n

; F ) of all n-linear mappings of finite type. It is plain that multilinear mappings of finite type are p-semi-integral, that is, L

f

(E

₁

, . . . , E

_n

; F ) ⊂ L

si,p

(E

₁

, . . . , E

_n

; F ). It is clear that to each operator in L

f

(E

₁

, . . . , E

_n

; F ) corresponds a tensor in E

^₁

⊗ · · · ⊗ E

^n

⊗ F via the canonical mapping

u =



m j=1

λ

j

ϕ

_1,j

⊗ · · · ⊗ ϕ

n,j

⊗ b

j

−→ T

u

=



m j=1

λ

j

ϕ

_1,j

× · · · × ϕ

n,j

b

j

, (3)

where λ

j

∈ K, ϕ

l,j

∈ E

^_l

, l = 1, . . . , n, b

j

∈ F , j = 1, . . . , m. Next we will see that, in some cases, these mappings are isometries.

12 Proposition. Let E

₁

, . . . , E

n

and F be Banach spaces and let p ≥ 1.

Given T ∈ L

f

(E

₁

, . . . , E

n

; F ), define

T 

f,p

:= inf (λ

j

)

^m_j=1



q

⎛

⎜ ⎜

⎜ ⎝ sup

φ_l∈B_E

l=1,...,nl



m j=1

|φ

1

(ϕ

_1,j

) · · · φ

n

(ϕ

n,j

)|

^p

⎞

⎟ ⎟

⎟ ⎠

1/p

(b

j

)

^m_j=1



∞

where the infimum is taken over all representations of T as in (2), and q ≥ 1 with

¹_p

+

¹_q

= 1.

Then  · 

f,p

is a norm on L

f

(E

₁

, . . . , E

n

; F ) with the following properties : (a) For every u ∈ E

^₁

⊗ · · · ⊗ E

n^

⊗ F we have that T

u

 ≤ T

u



f,p

= σ

p

(u).

Consequently, (L

f

(E

₁

, . . . , E

n

; F ),  . 

f,p

) is isometrically isomorphic to (E

^₁

⊗ · · · ⊗ E

n^

⊗ F, σ

p

) via the mapping given in (3).

(b) For every ϕ

_l

∈ E

_l^

, l = 1, . . . , n, and b ∈ F we have that ϕ

1

× · · · × ϕ

_n

b 

f,p

= ϕ

1

 · · · ϕ

n

 · b.

Proof. Following the lines of the proof of Proposition 2 it is easy to see that  · 

f,p

is a norm on L

f

(E

₁

, . . . , E

n

; F ).

(a) Since the equality T

u



f,p

= σ

p

(u) is trivial we show that T

u

 ≤ T

u



f,p

. Given x

_l

∈ E

l

with x

_l

= 0, l = 1, . . . , n, by H¨older’s inequality we have

T

u

(x

₁

, . . . , x

n

) 

^p

≤ x

1



^p

· · · x

n



^p

(b

j

)

^m_j=1



^p_∞

(λ

j

)

^m_j=1



^pq

sup

φ_l∈B_E

l=1,...,nl



m j=1

|φ

1

(ϕ

_1,j

) · · · φ

n

(ϕ

n,j

)|

^p

.

So, it follows that T

u

(x

₁

, . . . , x

n

) ≤ T

u



f,p

x

1

 · · · x

n

 and we have (a).

(b) Take ϕ

_l

∈ E

_l^

, l = 1, . . . , n, and b ∈ F . It is immediate that ϕ

1

× · · · × ϕ

_n

b 

f,p

≤ ϕ

1

 · · · ϕ

n

 · b. To prove the reverse inequality we use (a).

For every x

_l

∈ E

l

, l = 1, . . . , n, we have

|ϕ

1

(x

₁

) | · · · |ϕ

n

(x

_n

) |b ≤ ϕ

1

× · · · × ϕ

n

b x

1

 · · · x

n



≤ ϕ

1

× · · · × ϕ

n

b

f,p

x

1

 · · · x

n

.

Taking the supremum over B

El

, l = 1, . . . , n, we see that ϕ

1

 · · · ϕ

n

 ·

b ≤ ϕ

1

× · · · × ϕ

n

b

f,p

.

QED

By Proposition 12(b) we see that ϕ

1

× · · · × ϕ

n

b 

f,p

= ϕ

1

× · · · × ϕ

n

b 

si,p

for every ϕ

l

∈ E

_l^

, l = 1, . . . , n, and every b ∈ F with p ≥ 1. We do not know if

T 

f,p

= T 

si,p

whenever T ∈ L

f

(E

₁

, . . . , E

n

; F ).

13 Remark. When E

₁

, . . . , E

n

are reflexive Banach spaces the norm  · 

f,p

on L

f

(E

₁

, . . . , E

n

; F ) reduces to the following equivalent formulation: Given T ∈ L

f

(E

₁

, . . . , E

n

; F ), we have that

T 

f,p

= inf (λ

j

)

^m_j=1



q

⎛

⎜ ⎜

⎝ sup

_x

l∈B_El l=1,...,n



m j=1

|ϕ

1,j

(x

₁

) · · · ϕ

n,j

(x

n

)|

^p

⎞

⎟ ⎟

⎠

1/p

(b

j

)

^m_j=1



∞

where the infimum is taken over all representations of T as in (2), and q ≥ 1 with

¹_p

+

¹_q

= 1.

Next result provides a relation between ( L

si,p

(E

₁^

, . . . , E

^_n

; F

^

),  · 

si,p

) and (L

f

(E

₁

, . . . , E

n

; F ),  · 

f,p

), which gives a predual of (L

si,p

(E

₁^

, . . . , E

_n^

; F

^

), ·



si,p

), and also shows another predual of (L

si,p

(E

₁

, . . . , E

n

; F

^

),  · 

si,p

) in case of E

₁

, . . . , E

n

being reflexive spaces.

14 Proposition. Let E

₁

, . . . , E

_n

be Banach spaces and let p ≥ 1.

(a) Then (L

si,p

(E

₁^

, . . . , E

_n^

; F

^

), · 

si,p

) is isometrically isomorphic to (L

f

(E

₁

, . . . , E

n

; F ),  · 

f,p

)

^

by the mapping

T (ψ)(ϕ

₁

, . . . ϕ

n

)(b) = ψ(ϕ

₁

× · · · × ϕ

n

b),

where b ∈ F , ϕ

l

∈ E

_l^

, l = 1, . . . , n, and ψ ∈ (L

f

(E

₁

, . . . , E

_n

; F ),  · 

f,p

)

^

. If, in addition, E

₁

, . . . , E

_n

are reflexive Banach spaces then

(b) ( L

si,p

(E

₁

, . . . , E

n

; F

^

), ·

si,p

) and ( L

f

(E

₁^

, . . . , E

_n^

; F ), ·

f,p

)

^

are isomet- ric via the mapping

T (ψ)(x

₁

, . . . x

_n

)(b) = ψ(x

₁

× · · · × x

n

b),

where b ∈ F , x

l

∈ E

l

, l = 1, . . . , n, and ψ ∈ (L

f

(E

₁^

, . . . , E

^_n

; F ),  · 

f,p

)

^

. Proof. (a) follows from Propositions 6 and 12 and (b) is a straightforward

consequence of (a)

^QED

In the next by combining the previous results and taking F = K, in partic- ular, we obtain the following.

15 Corollary. Let E

1

, . . . , E

n

be Banach spaces and let p ≥ 1. Then the following isometries hold true:

(a) ( L

si,p

(E

₁^

, . . . , E

_n^

), ·

si,p

) ∼ = (E

₁^

⊗· · ·⊗E

^n

; σ

p

)

^

∼ = (E

₁^

⊗· · ·⊗E

n^

⊗K; σ

p

)

^

∼ = ( L

f

(E

₁

, . . . , E

n

),  · 

f,p

)

^

.

If, in addition, E

₁

, . . . , E

n

are reflexive Banach spaces then the following

isometries hold true:

(b) (L

si,p

(E

₁

, . . . , E

n

), ·

si,p

) ∼ = (E

₁

⊗· · ·⊗E

n

; σ

p

)

^

∼ = (E

₁

⊗· · ·⊗E

n

⊗K; σ

p

)

^

∼ = (L

f

(E

₁^

, . . . , E

^_n

),  · 

f,p

)

^

.

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