Isomorphisms between spaces of multilinear mappings or homogeneous polynomials

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Note di Matematica 29, n. 2, 2009, 55–76.

Isomorphisms between spaces of multilinear mappings or homogeneous polynomials

Po Ling, Kuo

ⁱ

Departamento Acadêmico de Matemática, Universidade Tecnológica Federal do Paraná Av. Sete de Setembro, 3165, CEP 80230-901, Curitiba, PR, Brazil

kuopoling@gmail.com

Received: 22/08/2008; accepted: 13/02/2009.

Abstract. LetE and F be Banach spaces. Our objective in this work is to ﬁnd conditions under which, whenever the topological dual spacesEandFare isomorphic, the spaces of multilinear mappings (resp. homogeneous polynomials) onE and F are isomorphic as well.

We also examine the corresponding problem for the spaces of multilinear mappings (resp.

homogeneous polynomials) of a certain type, for instance of ﬁnite, nuclear, compact or weakly compact type.

Keywords:Banach space , Polynomials , Isomorphisms.

MSC 2000 classiﬁcation:primary 46G20, secondary 46G25

Notation

Throughout the whole paper D,E, F and G always denote Banach spaces over the same ﬁeld K, where K = R or C. N denotes the set of all positive integers. L(

^m

E; G) denotes the vector space of all continuous m-linear mappings from E

^m

into G. L(

^m

E; G) is a Banach space under its natural norm. If G = K, we write L(

^m

E; K) = L(

^m

E). If m = 1, we write L(

¹

E; G) = L(E; G). If m = 1 and G = K, we write L(E) = E

, the topological dual of E. The mapping

I

m

: L(

^m+n

E; G) → L(

^m

E; L(

ⁿ

E; G))

deﬁned by I

m

A(x)(y) = A(x, y) for all A ∈ L(

^m+n

E; G), x ∈ E

^m

, y ∈ E

ⁿ

, is an isometric isomorphism. Likewise the mapping

T

^t

: A ∈ L(

^m

E; L(

ⁿ

F ; G)) → A

^t

∈ L(

ⁿ

F ; L(

^m

E; G))

deﬁned by A

^t

(y)(x) = A(x)(y) for all A ∈ L(

^m

E; L(

ⁿ

F ; G)), x ∈ E

^m

, y ∈ F

ⁿ

, is an isometric isomorphism. Let L

^s

(

^m

E; G) denote the subspace of all A ∈ L(

^m

E; G) which are symmetric. Let P(

^m

E; G) denote the vector space of all continuous m − homogeneous polynomials from E into G. If G = K, we write

iThis work had ﬁnancial support from CNPq(Brazil).

Note Mat. 29 (2009), n. 2, 55-76 ISSN 1123-2536, e-ISSN 1590-0932 DOI 10.1285/i15900932v29n2p55

__________________________________________________________________

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P(

^m

E; K) = P(

^m

E). For each A ∈ L(

^m

E; G) let ' A ∈ P(

^m

E; G) be deﬁned by ' A(x) = Ax

^m

for every x ∈ E. The mapping A → ' A induces a topological isomorphism between L

^s

(

^m

E; G) and P(

^m

E; G).

1 Deﬁnition. A mapping A ∈ L(

^m

E; G) is said to be of ﬁnite type if there exist c

₁

, . . . , c

_n

∈ G, ϕ

1i

, . . . , ϕ

_mi

∈ E

, 1 ≤ i ≤ n such that A can be written in the form: A(x

₁

, . . . , x

_m

) =

_n

i=1

ϕ

_1i

(x

₁

) ···ϕ

mi

(x

_m

)c

_i

for all (x

₁

, . . . , x

_m

) ∈ E

^m

. 2 Deﬁnition. A polynomial P ∈ P(

^m

E; G) is said to be of ﬁnite type, if there exist c

₁

, . . . , c

_n

∈ G, ϕ

1

, . . . , ϕ

_n

∈ E

such that P can be written of the form: P (x) =

_n

i=1

ϕ

i

(x)

^m

c

i

for all x ∈ E.

3 Deﬁnition. A mapping A ∈ L(

^m

E; G) is said to be nuclear, if there exist sequence (ϕ

ji

)

_i∈N

in E

, 1 ≤ j ≤ m and (c

i

)

_i∈N

in G with

_∞

i=1

ϕ

1i

· · ·

ϕ

mi

c

i

< ∞ such that A(x

1

, . . . , x

m

) =

_∞

i=1

ϕ

_1i

(x

₁

) · · · ϕ

mi

(x

m

)c

i

for all (x

₁

, . . . , x

m

) ∈ E

^m

.

4 Deﬁnition. A polynomial P ∈ P(

^m

E; G) is said to be nuclear, if there exist sequences (ϕ

i

)

_i∈N

in E

, and (c

i

)

_i∈N

in G with

_∞

i=1

ϕ

i

^m

c

i

< ∞ such that P (x) =

_∞

i=1

ϕ

i

(x)

^m

c

i

for all x ∈ E.

Let L

_f

(

^m

E; G) denote the space of all A ∈ L(

^m

E; G) which are of ﬁnite type. Let P

f

(

^m

E; G) denote the space of all P ∈ P(

^m

E; G) which are of ﬁnite type. Let L

_N

(

^m

E; G) denote the space of all A ∈ L(

^m

E; G) which are nuclear endowed with the nuclear norm A

N

= inf

_∞

i=1

ϕ

1i

· · · ϕ

mi

c

i

, where the inﬁmum is taken over all sequences (ϕ

_ji

)

_i∈N

and (c

_i

)

_i∈N

which satisfy the deﬁnition. When G = K, we write L

Θ

(

^m

E, K) = L

Θ

(

^m

E), where Θ = f or N . Let P

N

(

^m

E; G) denote the space of all P ∈ P(

^m

E; G) which are nuclear, endowed with the nuclear norm P

N

= inf

_∞

i=1

ϕ

i

^m

c

i

, where the inﬁmum

is taken over all sequences (ϕ

i

)

_i∈N

and (c

i

)

_i∈N

which satisfy the deﬁnition. When

G = K, we write P

Θ

(

^m

E, K) = P

Θ

(

^m

E), where Θ = f or N . Let us recall that

A ∈ L(

^m

E; G) is a compact (resp. weakly compact) mapping if A(B

E^m

) is

relatively compact in G (resp. for the weak topology), where B

E^m

denotes the

closed unit ball of E

^m

. Let L

K

(

^m

E; G) ( resp. L

W K

(

^m

E; G) ) denote the space

of all A ∈ L(

^m

E; G) which are compact (resp. weakly compact). Recall that

P ∈ P(

^m

E; G) is a compact (resp. weakly compact) polynomial if P (B

E

) is

relatively compact in G (resp. for the weak topology), where B

E

denotes the

closed unit ball of E. Let P

K

(

^m

E; G) (resp. P

W K

(

^m

E; G)) denote the space

of all compact (resp. weakly compact) homogeneous polynomials from E into

G. We observe that in the case where G = K, all the continuous homogeneous

polynomials are compact (resp. weakly compact). For background information

on multilinear mappings and homogeneous polynomials we refer to the books

[7] and [12].

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1 Isomorphisms between spaces of multilinear mappings or homogeneous polynomials

In this work, we use the Nicodemi sequences deﬁned in [8] to prove all the theorems. We recall the deﬁnition of the Nicodemi sequences.

5 Deﬁnition. [8] Given a continuous linear operator R

₁

: L(E; G) −→

L(F ; G), let R

m

: L(

^m

E; G) −→ L(

^m

F ; G) be inductively deﬁned by R

_m+1

A = I

_m⁻¹

[R

m

◦ (R

1

◦ I

m

(A))

^t

]

^t

for all A ∈ L(

^m+1

E; G) and m ∈ N.

6 Example. [8, Example 1.2] Let T

₁

: E

→ E

be the natural embedding and let T

_m

: L(

^m

E) → L(

^m

E

) be the Nicodemi sequence of operators beginning with T

₁

. This sequence is precisely the sequence of operators constructed by Aron and Berner in [1, Proposition 2.1].

7 Deﬁnition. [8, Proposition 4.1] Given R

₁

∈ L(E

; F

), let R

₁

∈ L(L(E; G

), L(F ; G

)) be deﬁned by R

₁

A(y)(z) = R

₁

(δ

z

◦ A)(y) for all A ∈ L(E; G

), y ∈ F and z ∈ G, where δ

z

: G

→ K is deﬁned by δ

z

(z

) = z

(z) for all z

∈ G

. If (R

m

) and (( R

m

) are the corresponding Nicodemi sequences, then R (

m

A(y)(z) = R

m

(δ

z

◦ A)(y) for all A ∈ L(

^m

E; G

), y ∈ F

^m

and z ∈ G.

8 Example. Let T

m

: L(

^m

E) −→ L(

^m

E

) be the Nicodemi sequence of operators beginning with the natural embedding T

₁

: E

→ E

, and let T

m

: L(

^m

E; G

) −→ L(

^m

E

; G

) be the corresponding sequence for vector- valued multilinear mappings (See [8, Proposition 4.1]). We observe that as G

is a C

1

−space, the sequence T

m

: L(

^m

E; G

) −→ L(

^m

E

; G

) ⊂ L(

^m

E

; G

) coincides with the sequence of operators constructed by Aron and Berner in [1, Proposition 2.1].

The next theorem shows the relationship between an arbitrary Nicodemi sequence of scalar-valued mappings and the Nicodemi sequence beginning with the natural embedding E

→ E

.

9 Theorem. Let R

_m

: L(

^m

E) −→ L(

^m

F ) be a Nicodemi sequence, let T

_m

: L(

^m

E) −→ L(

^m

E

) be the Nicodemi sequence beginning with the natural embedding T

₁

= J

_E

: E

→ E

, and let J

_F

: F → F

be the natural embedding.

Then

R

m

A(y

₁

, . . . , y

m

) = T

m

A(R

₁

(J

F

y

₁

), . . . , R

₁

(J

F

y

m

)) for all A ∈ L(

^m

E), and y

₁

, . . . , y

m

∈ F , where R

₁

is the transpose of R

₁

.

Proof. We will prove this theorem by induction on m. If A ∈ L(E) = E

, we have

R

₁

A(y) = J

F

y, R

₁

A = R

₁

(J

F

y), A = T

1

A, R

₁

(J

F

y) = T

1

A(R

₁

(J

F

y))

(4)

for all y ∈ F. Now let us assume that the identity in Theorem 9 is true for m−

linear forms. Let A ∈ L(

^m+1

E). We will prove that

T

_m+1

A(z

₁

, . . . , z

_m

, R

₁

(J

_F

y)) = T

_m

[(R

₁

◦ I

m

A)

^t

(y)](z

₁

, . . . , z

_m

) (1) for all z

₁

, . . . , z

m

∈ E

and y ∈ F . From the deﬁnition of Nicodemi sequences, it follows that

T

_m+1

A(z

₁

, . . . , z

m

, R

₁

(J

F

y)) = T

m

[(T

₁

◦ I

m

A)

^t

(R

₁

(J

F

y))](z

₁

, . . . , z

m

). (2) Therefore, to get (1) comparing with (2), it is enough to prove that

(R

₁

◦ I

m

A)

^t

(y) = (T

₁

◦ I

m

A)

^t

(R

₁

(J

F

y)).

In fact,

(T

₁

◦ I

m

A)

^t

(R

₁

(J

F

y))(x

₁

, . . . , x

m

) = T

₁

[I

m

A(x

₁

, . . . , x

m

)](R

₁

(J

F

y))

= R

₁

(J

_F

y), I

_m

A(x

₁

, . . . , x

_m

)

= J

F

y, R

₁

[I

m

A(x

₁

, . . . , x

m

)]

= R

1

[I

_m

A(x

₁

, . . . , x

_m

)], y

= R

₁

[I

m

A(x

₁

, . . . , x

m

)](y)

= (R

₁

◦ I

m

A)

^t

(y)(x

₁

, . . . , x

_m

)

for all x

₁

, . . . , x

m

∈ E. Thus by the induction hypothesis and (1) it follows that for all y

₁

, . . . , y

_m+1

∈ F

R

_m+1

A(y

₁

, . . . , y

_m+1

) =R

_m

[(R

₁

◦ I

m

A)

^t

(y

_m+1

)](y

₁

, . . . , y

_m

)

=T

m

[(R

₁

◦ I

m

A)

^t

(y

_m+1

)](R

₁

(J

F

y

₁

), . . . , R

₁

(J

F

y

m

))

=T

_m+1

A(R

₁

(J

_F

y

₁

), . . . , R

₁

(J

_F

y

_m

), R

₁

(J

_F

y

_m+1

)).

QED

We next extend Theorem 9 to the case of a Nicodemi sequence of vector- valued mappings. Let us recall that each Nicodemi sequence (R

_m

) for scalar- valued mappings yields a Nicodemi sequence (( R

m

) for vector - valued mappings.

10 Theorem. Let R

_m

: L(

^m

E) −→ L(

^m

F ) be a Nicodemi sequence, and let T

_m

: L(

^m

E) −→ L(

^m

E

) be the Nicodemi sequence beginning with the natural embedding T

₁

= J

_E

: E

→ E

. Let

( R

m

: L(

^m

E; G

) → L(

^m

F ; G

)

(5)

and T

m

: L(

^m

E; G

) → L(

^m

E

; G

)

be the corresponding Nicodemi sequences for vector-valued multilinear mappings.

Then ( R

m

A(y

₁

, . . . , y

m

) = T

m

A(R

₁

(J

F

y

₁

), . . . , R

₁

(J

F

y

m

))

for all A ∈ L(

^m

E; G

), and y

₁

, . . . , y

m

∈ F , where R

₁

is the transpose of R

₁

. Proof. We will prove this theorem by induction on m. If A ∈ L(E; G

), it follows that

R

₁

A(y)(z) =R

₁

(δ

z

◦ A)(y)

= J

F

y, R

₁

(δ

_z

◦ A)

= R

₁

(J

F

y), (δ

z

◦ A)

= T

1

(δ

_z

◦ A), R

₁

(J

_F

y)

= T

₁

A(R

₁

(J

F

y))(z)

for all y ∈ F and z ∈ G. Now let us assume that the identity is true for m−

linear forms. Let A ∈ L(

^m+1

E; G

). We prove initially that

T

_m+1

A(z

₁

, . . . , z

m

, R

₁

(J

F

y)) = T

m

[( R

₁

◦ I

m

A)

^t

(y)](z

₁

, . . . , z

m

) (3) for all z

₁

, . . . , z

m

∈ E

and y ∈ F . From the deﬁnition of Nicodemi sequences, it follows that

T

_m+1

A(z

₁

, . . . , z

_m

, R

₁

(J

_F

y)) = T

_m

[( T

₁

◦ I

m

A)

^t

(R

₁

(J

_F

y))](z

₁

, . . . , z

_m

). (4)

Therefore, to get (3) comparing with (4), it is enough to prove that ( R

₁

◦ I

m

A)

^t

(y) = ( T

₁

◦ I

m

A)

^t

(R

₁

(J

F

y)).

In fact,

( T

₁

◦ I

m

A)

^t

(R

₁

(J

_F

y))(x

₁

, . . . , x

m

)(z) = T

₁

[I

m

A(x

₁

, . . . , x

m

)](R

₁

(J

_F

y))(z)

=T

₁

[δ

z

◦ I

m

A(x

₁

, . . . , x

m

)](R

₁

(J

F

y))

= R

₁

(J

_F

y), δ

z

◦ I

m

A(x

₁

, . . . , x

m

)

= J

F

y, R

₁

[δ

z

◦ I

m

A(x

₁

, . . . , x

m

)]

= R

1

[δ

z

◦ I

m

A(x

₁

, . . . , x

m

)], y

=R

₁

[δ

z

◦ I

m

A(x

₁

, . . . , x

m

)](y)

=( R

₁

◦ I

m

A)

^t

(y)(x

₁

, . . . , x

m

)(z)

(6)

for all x

₁

, . . . , x

m

∈ E and z ∈ G. Thus, from the induction hypothesis and (3), it follows that for all y

₁

, . . . , y

_m+1

∈ F

R

_m+1

A(y

₁

, . . . , y

_m+1

) =( R

m

[( R

₁

◦ I

m

A)

^t

(y

_m+1

)](y

₁

, . . . , y

m

)

= T

_m

[( R

₁

◦ I

m

A)

^t

(y

_m+1

)](R

₁

(J

_F

y

₁

), . . . , R

₁

(J

_F

y

_m

))

= T

_m+1

A(R

₁

(J

_F

y

₁

), . . . , R

₁

(J

_F

y

_m

), R

₁

(J

_F

y

_m+1

)).

QED

Recall that a Banach space E is said to be Arens - regular if all linear operator E −→ E

are weakly compact, and symmetrically Arens - regular if this is so for all symmetric linear operators. An operator T : E → E

is said to be symmetric if T x(y) = T y(x) for all x, y ∈ E(see [3] and [9]). Let us recall that if E is symmetrically Arens - regular, then T

m

A ∈ L

^s

(

^m

E

) for all A ∈ L

^s

(

^m

E), where (T

m

) is the Nicodemi sequence beginning with the natural embedding T

₁

= J

E

: E

→ E

(see [2, Theorem 8.3]). Now, we are ready to study the theorems of preservation of symmetric multilinear mappings.

11 Theorem. Let R

m

: L(

^m

E) −→ L(

^m

F ) be a Nicodemi sequence and let T

m

: L(

^m

E) −→ L(

^m

E

) be the Nicodemi sequence beginning with the natural embedding T

₁

= J

_E

: E

→ E

. If T

m

A is symmetric, then R

m

A is also symmetric. In particular, if E is symmetrically Arens - regular, then R

m

A ∈ L

^s

(

^m

F ) for all A ∈ L

^s

(

^m

E).

Proof. By Theorem 9 we have that

R

m

A(y

₁

, . . . , y

m

) = T

m

A(R

₁

(J

F

y

₁

), . . . , R

₁

(J

F

y

m

)).

Thus, if T

m

A is symmetric, then R

m

A is also symmetric. Now if E is symmetrically Arens - regular, we have that the Aron - Berner extension T

m

A is symmetric for all A ∈ L

^s

(

^m

E)(See [2, Proposition 8.3]). We conclude that

R

m

A ∈ L

^s

(

^m

F ) for all A ∈ L

^s

(

^m

E).

^QED

12 Theorem. Let R

m

: L(

^m

E) −→ L(

^m

F ) be a Nicodemi sequence and let ( R

_m

: L(

^m

E; G

) −→ L(

^m

F ; G

) be the corresponding Nicodemi sequence for vector-valued multilinear mappings. If E is symmetrically Arens - regular, then ( R

_m

A ∈ L

^s

(

^m

F ; G

) for all A ∈ L

^s

(

^m

E; G

).

Proof. By [8, Proposition 4.1], we have that

( R

_m

A(y)(z) = R

_m

(δ

_z

◦ A)(y) (5)

for all A ∈ L(

^m

E, G

), y ∈ F

^m

and z ∈ G. The identity (5) and Theorem 11

imply that if A is symmetric, then ( R

m

A is also symmetric.

^QED

(7)

In Theorem 13 we will denote J

F

(y) by y for all y ∈ F and J

E

(x) by x for all x ∈ E.

13 Theorem. Let R

_m

: L(

^m

E) −→ L(

^m

F ) be a Nicodemi sequence, let T

_m

: L(

^m

E) −→ L(

^m

E

) be the Nicodemi sequence beginning with the natural embedding T

₁

= J

_E

: E

→ E

, and let Q

_m

: L(

^m

F ) −→ L(

^m

F

) be the Nicodemi sequence beginning with the natural embedding Q

₁

= J

_F

: F

→ F

. If T

_m

A is symmetric, then

Q

_m

◦ R

m

A(w

₁

, . . . , w

_m

) = T

_m

A(R

₁

w

₁

, . . . , R

₁

w

_m

) for all A ∈ L(

^m

E), w

j

∈ F

, j = 1, . . . , m, and m ∈ N.

Proof. We will prove by induction on k ∈ N that Q

m

◦ R

m

A(w

₁

, . . . , w

_k

, y

_k+1

, . . . , y

m

) =

T

m

A(R

₁

(w

₁

), . . . , R

₁

(w

k

), R

₁

(y

_k+1

), . . . , R

₁

(y

m

)) for all y

j

∈ F and w

j

∈ F

. Recall that (i) Q

m

B and T

m

A are weak

continuous in its ﬁrst variable for all B ∈ L(

^m

F ) and A ∈ L(

^m

E) by [8, Proposition 5.1]; (ii) elements of F

and of J

F

(F ) commute in variables of Q

m

B by [8, Lemma 3.4];

(iii) R

₁

is σ(F

, F

)−σ(E

, E

) continuous; (iv) by [8, Proposition 2.1] , we have that T

m

A(x

₁

, . . . , x

m

) = A(x

₁

, . . . , x

m

) for all A ∈ L(

^m

E) and x

₁

, . . . , x

m

∈ E;

and Q

m

B(y

₁

, . . . , y

m

) = B(y

₁

, . . . , y

m

) for all B ∈ L(

^m

F ) and y

₁

, . . . , y

m

∈ F . If k = 1, by Goldstine´ s theorem, there is a net (y

α

) ⊂ F such that y

α

−→ w

1

for the topology σ(F

, F

). Then by Theorem 9

Q

m

◦ R

m

A(w

₁

, y

₂

. . . , y

m

) =Q

m

(R

m

A)(w

₁

, y

₂

. . . , y

m

)

= lim

α

Q

_m

(R

_m

A)(y

_α

, y

₂

. . . , y

_m

)

= lim

α

(R

_m

A)(y

_α

, y

₂

. . . , y

_m

)

= lim

α

T

m

A(R

₁

y

α

, R

₁

y

₂

, . . . , R

₁

y

m

)

=T

m

A(R

₁

w

₁

, R

₁

y

₂

, . . . , R

₁

y

m

).

Now assuming that the identity holds for k, we will prove that the identity holds for k + 1. By the induction hypothesis and T

m

A being symmetric, it follows that Q

_m

◦ R

m

A(w

₁

, . . . , w

_k+1

,y

_k+2

, . . . , y

_m

)

=Q

m

(R

m

A)(w

₁

, . . . , w

_k+1

, y

_k+2

, . . . , y

m

)

= lim

α

Q

_m

(R

_m

A)(y

_α

, w

₂

, . . . , w

_k+1

, y

_k+2

, . . . , y

_m

)

= lim

α

Q

_m

(R

_m

A)(w

₂

, . . . , w

_k+1

, y

_α

, y

_k+2

, . . . , y

_m

)

(8)

= lim

α

T

m

A(R

₁

w

₂

, . . . , R

₁

w

_k+1

, R

₁

y

α

, R

₁

y

_k+2

, . . . , R

₁

y

m

)

= lim

α

T

m

A(R

₁

y

α

, R

₁

w

₂

, . . . , R

₁

w

_k+1

, R

₁

y

_k+2

, . . . , R

₁

y

m

)

=T

m

A(R

₁

w

₁

, . . . , R

₁

w

_k+1

, R

₁

y

_k+2

, . . . , R

₁

y

m

).

QED

Next we will prove that each isomorphism between E

and F

induces an isomorphism between L(

^m

E; G

) and L(

^m

F ; G

) for all m ∈ N. If E and F are symmetrically Arens - regular, then each isomorphism between E

and F

induces an isomorphism between P(

^m

E; G

) and P(

^m

F ; G

) for all m ∈ N.

Given a continuous linear operator

R

m

: L(

^m

E; G) −→ L(

^m

F ; G), we deﬁne

U

_m

: A ∈ L(

ⁿ

D; L(

^m

E; G)) −→ R

m

◦ A ∈ L(

ⁿ

D; L(

^m

F ; G)).

We observe that if R

_m

is an isomorphism then U

_m

is also an isomorphism, whose inverse

U

_m⁻¹

: L(

ⁿ

D; L(

^m

F ; G)) −→ L(

ⁿ

D; L(

^m

E; G))

is deﬁned by U

_m⁻¹

(B) = R

⁻¹_m

◦ B for all B ∈ L(

ⁿ

D; L(

^m

F ; G)) where R

⁻¹_m

is the inverse of R

m

. Thus, with the previous notations, it is possible to rewrite the deﬁnition of the Nicodemi operators in the following way:

14 Lemma. Given an operator

R

m

: L(

^m

E; G) −→ L(

^m

F ; G), the operator

R

_m+1

: L(

^m+1

E; G) −→ L(

^m+1

F ; G) is given by

R

_m+1

(A) = I

_m⁻¹

[R

_m

◦ (R

1

◦ I

m

(A))

^t

]

^t

= I

_m⁻¹

◦ T

^t

◦ U

m

◦ T

^t

◦ U

1

◦ I

m

(A) for all A ∈ L(

^m+1

E; G).

The following theorem was obtained in [4]. But we will need our proof in the proof of the subsequent theorem.

15 Theorem. If E

and F

are isomorphic, then L(

^m

E) and L(

^m

F ) are

isomorphic for all m ∈ N.

(9)

Proof. Since E

and F

are isomorphic, there exists an isomorphism R

₁

: E

−→ F

. Let R

m

: L(

^m

E) −→ L(

^m

F ) be the Nicodemi sequence beginning with R

₁

. We will prove by induction on m ∈ N that R

m

is an isomorphism between L(

^m

E) and L(

^m

F ) for all m ∈ N. By hypothesis R

1

: E

−→ F

is an isomorphism. Assuming that R

₁

and R

m

are isomorphisms, we show that R

_m+1

is also an isomorphism. In fact, by Lemma 1 it is possible to rewrite

R

_m+1

= I

_m⁻¹

◦ T

^t

◦ U

m

◦ T

^t

◦ U

1

◦ I

m

.

Since R

₁

and R

m

are isomorphisms, we have that U

₁

and U

m

are also isomorphisms. Thus R

_m+1

is an isomorphism between L(

^m+1

E) and L(

^m+1

F ) being a composite of isomorphisms. Therefore L(

^m

E) and L(

^m

F ) are isomorphic for

all m ∈ N.

^QED

16 Theorem. If E

and F

are isomorphic, then L(

^m

E; G

) and L(

^m

F ; G

) are isomorphic for all m ∈ N.

Proof. Since E

and F

are isomorphic, there exists an isomorphism R

₁

: E

−→ F

. Let R

_m

: L(

^m

E) −→ L(

^m

F ) be the Nicodemi sequence beginning with R

₁

. Let

R (

_m

: L(

^m

E, G

) −→ L(

^m

F, G

)

be the corresponding Nicodemi sequence for vector-valued multilinear mappings.

We observe that

(δ

_z

◦ ( R

_m

A) = R

_m

(δ

_z

◦ A) (6)

for all z ∈ G and A ∈ L(

^m

E; G

). In fact, by [8, Proposition 4.1],

(δ

z

◦ ( R

m

A)(y) = ( R

m

A(y)(z) = R

m

(δ

z

◦ A)(y)

for all y ∈ F

^m

. It follows from the proof of Theorem 15 that R

m

is an isomorphism for all m ∈ N. Let S

m

: L(

^m

F ) −→ L(

^m

E) denote the inverse of R

m

for all m ∈ N. We deﬁne

S

m

: L(

^m

F ; G

) −→ L(

^m

E; G

)

by S

m

B(x)(z) = S

m

(δ

z

◦B)(x) for all B ∈ L(

^m

F ; G

), x ∈ E

^m

, z ∈ G and m ∈ N.

Thus S

m

is linear and continuous. We will show that ( R

m

is an isomorphism

(10)

between L(

^m

E; G

) and L(

^m

F ; G

) for all m ∈ N. In fact, we have that by (6) S

m

◦ ( R

m

A(x)(z) = S

m

(( R

m

A)(x)(z)

=S

m

(δ

_z

◦ ( R

_m

A)(x)

=S

m

(R

m

(δ

z

◦ A))(x)

=[S

_m

◦ R

m

(δ

_z

◦ A)](x)

=(δ

z

◦ A)(x)

=A(x)(z)

for all A ∈ L(

^m

E, G

), x ∈ E

^m

and z ∈ G. In a similar way, we can get that (( R

m

◦ S

m

)B = B for all B ∈ L(

^m

F ; G

).

^QED

17 Theorem. If E and F are symmetrically Arens - regular, and E

and F

are isomorphic, then L

^s

(

^m

E) and L

^s

(

^m

F ) are isomorphic for all m ∈ N.

Proof. We write J

_E

(x) = x for all x ∈ E. Recall that by [8, Proposition 2.1] T

_m

A(x

₁

, . . . , x

_m

) = A(x

₁

, . . . , x

_m

) for all A ∈ L(

^m

E) and x

₁

, . . . , x

_m

∈ E.

Since E

and F

are isomorphic, there exists an isomorphism R

₁

: E

−→ F

. Let R

_m

: L(

^m

E) −→ L(

^m

F ) be the Nicodemi sequence beginning with R

₁

. Let S

₁

= R

⁻¹₁

: F

−→ E

be the inverse of R

₁

and let S

_m

: L(

^m

F ) −→ L(

^m

E) be the Nicodemi sequence beginning with S

₁

. Since F is symmetrically Arens - regular, we have by Theorem 11 that S

_m

(L

^s

(

^m

F )) ⊂ L

^s

(

^m

E). By Theorem 9 we have that S

_m

B(x

₁

, . . . , x

_m

) = Q

_m

B(S

₁

x

₁

, . . . , S

₁

x

_m

) for all B ∈ L(

^m

F ), and x

₁

, . . . , x

_m

∈ E, where S

₁

is the transpose of S

₁

. In particular, we have that S

_m

(R

_m

A)(x

₁

, . . . , x

_m

) = Q

_m

(R

_m

A)(S

₁

x

₁

, . . . , S

₁

x

_m

) (7) for all A ∈ L(

^m

E). On the other hand, since E is symmetrically Arens - regular, it follows from Theorem 11 that R

_m

(L

^s

(

^m

E)) ⊂ L

^s

(

^m

F ). Moreover, by Theorem 13 we have that

Q

_m

(R

_m

A)(S

₁

x

₁

, . . . , S

₁

x

_m

) = T

_m

A(R

₁

(S

₁

x

₁

), . . . , R

₁

(S

₁

x

_m

)), (8) for all A ∈ L

^s

(

^m

E). Therefore we conclude by (7) and (8) that

S

_m

(R

_m

A)(x

₁

, . . . , x

_m

) =Q

_m

(R

_m

A)(S

₁

x

₁

, . . . , S

₁

x

_m

)

=T

m

A(R

₁

(S

₁

x

₁

), . . . , R

₁

(S

₁

x

m

))

=T

_m

A(x

₁

, . . . , x

_m

)

=A(x

₁

, . . . , x

m

), for all A ∈ L

^s

(

^m

E), that is

(S

m

◦ R

m

)A = A (9)

(11)

for all A ∈ L

^s

(

^m

E). In an analogous way, we can prove that (R

m

◦ S

m

)B = B

for all B ∈ L

^s

(

^m

F ).

^QED

The following Corollary 18 was proven by Lassalle - Zalduendo in [11] and by F. Cabello S´ anchez, J. Castillo and R. Garc´ıa in [4] by a diﬀerent method.

18 Corollary. If E and F are symmetrically Arens - regular, and E

and F

are isomorphic, then P(

^m

E) and P(

^m

F ) are isomorphic for all m ∈ N.

19 Theorem. If E and F are symmetrically Arens - regular, and E

and F

are isomorphic, then L

^s

(

^m

E; G

) and L

^s

(

^m

F ; G

) are isomorphic for all m ∈ N.

Proof. Since E

and F

are isomorphic, there exists an isomorphism R

₁

: E

−→ F

. Let R

_m

: L(

^m

E) −→ L(

^m

F ) be the Nicodemi sequence beginning with R

₁

. Let S

₁

= R

⁻¹₁

: F

−→ E

be the inverse of R

₁

, and let S

_m

: L(

^m

F ) −→

L(

^m

E) be the Nicodemi sequence beginning with S

₁

. Let ( R

m

: L(

^m

E; G

) −→

L(

^m

F ; G

) and S

_m

: L(

^m

F ; G

) −→ L(

^m

E; G

) be the corresponding Nicodemi sequence for vector-valued multilinear mappings. By Theorem 12 we have that R (

m

(L

^s

(

^m

E; G

)) ⊂ L

^s

(

^m

F ; G

). (10)

and

S

m

(L

^s

(

^m

F ; G

)) ⊂ L

^s

(

^m

E; G

). (11) It follows from the proof of Theorem 16 that ( R

m

is an isomorphism between L(

^m

E; G

) and L(

^m

F ; G

) such that ( R

m

−1

= S

m

. By (10) and (11), we have that R (

m

|

L^s(^mE;G)

is an isomorphism between L

^s

(

^m

E; G

) and L

^s

(

^m

F ; G

).

^QED

The following Corollary 20 was proven for Carando - Lassalle in [5] by a diﬀerent method.

20 Corollary. If E and F are symmetrically Arens - regular, and E

and F

are isomorphic, then P(

^m

E; G

) and P(

^m

F ; G

) are isomorphic for all m ∈ N.

2 Isomorphisms between spaces of nuclear multilinear mappings or homogeneous polynomials.

We will use the following notation:

(ϕ

₁

⊗ ϕ

2

⊗ c)(x

1

, x

₂

) = ϕ

₁

(x

₁

)ϕ

₂

(x

₂

)c

for all ϕ

₁

, ϕ

₂

∈ E

, c ∈ F and x

1

, x

₂

∈ E. The next lemma comes essentially

from Aron and Berner [1].

(12)

21 Lemma. [1, Proposition 2.2] Let T

m

: L(

^m

E) −→ L(

^m

E

) be the Nicodemi sequence beginning with the natural embedding T

₁

= J

E

: E

→ E

. Let A ∈ L

f

(

^m

E) and let c

₁

, . . . , c

n

∈ K, ϕ

1i

, . . . , ϕ

mi

∈ E

, 1 ≤ i ≤ n such that

A(x

₁

, . . . , x

_m

) =

n i=1

ϕ

_1i

(x

₁

) · · · ϕ

mi

(x

_m

)c

_i

for all (x

₁

, . . . , x

m

) ∈ E

^m

, then

T

m

(A)(x

₁

, . . . , x

_m

) =

n i=1

(T

₁

ϕ

_1i

)(x

₁

) · · · (T

1

ϕ

mi

)(x

_m

)c

i

for all (x

₁

, . . . , x

_m

) ∈ (E

)

^m

. In particular T

m

(A) ∈ L

f

(

^m

E

).

22 Lemma. Let T

_m

: L(

^m

E) −→ L(

^m

E

) be the Nicodemi sequence beginning with the natural embedding T

₁

= J

_E

: E

→ E

. Let A ∈ L

N

(

^m

E) and let (ϕ

_ji

)

_i∈N

in E

, 1 ≤ j ≤ m and (c

i

)

_i∈N

in K with

∞ i=1

ϕ

1i

· · · ϕ

mi

c

i

< ∞

such that

A(x

₁

, . . . , x

_m

) =

∞ i=1

ϕ

_1i

(x

₁

) · · · ϕ

mi

(x

_m

)c

_i

for all (x

₁

, . . . , x

m

) ∈ E

^m

. Then T

m

(A) has the following form:

T

m

(A)(x

₁

, . . . , x

_m

) =

∞ i=1

(T

₁

ϕ

_1i

)(x

₁

) · · · (T

1

ϕ

mi

)(x

_m

)c

i

for all (x

₁

, . . . , x

_m

) ∈ (E

)

^m

. In particular T

m

(A) ∈ L

N

(

^m

E

).

Proof. Let A

_n

=

_n

i=1

ϕ

_1i

⊗···⊗ϕ

mi

⊗c

i

for all n ∈ N. Then A

n

∈ L

f

(

^m

E) and A

_n

→ A for the nuclear norm, that is

A

n

− A

N

−→ 0. (12)

By Lemma 21 T

_m

A

_n

has the following form: T

_m

A

_n

=

_n

i=1

T

₁

ϕ

_1i

⊗···⊗T

1

ϕ

_mi

⊗c

i

for all n ∈ N. It follows that T

m

A

_n

−→

_∞

i=1

T

₁

ϕ

_1i

⊗ · · · ⊗ T

1

ϕ

_mi

⊗ c

i

for the nuclear norm. In particular

T

m

A

n

−→

∞ i=1

T

₁

ϕ

_1i

⊗ · · · ⊗ T

1

ϕ

mi

⊗ c

i

(13)

pointwise. On the other hand, we have that

T

m

A

n

→ T

m

A (14)

pointwise. Then by (12)

T

m

A(x

₁

, . . . , x

_m

) − T

m

A

_n

(x

₁

, . . . , x

_m

)

=(T

m

A − T

m

A

n

)(x

₁

, . . . , x

_m

)

= T

m

(A − A

n

)(x

₁

, . . . , x

_m

)

≤T

m

A − A

n

(x

₁

, . . . , x

_m

)

≤T

m

A − A

n

N

(x

₁

, . . . , x

_m

) −→ 0 for all x

₁

, . . . , x

_m

∈ E

. By (13) and (14), we have that

T

m

(A)(x

₁

, . . . , x

_m

) =

∞ i=1

(T

₁

ϕ

_1i

)(x

₁

) · · · (T

1

ϕ

mi

)(x

_m

)c

i

for all x

₁

, . . . , x

_m

∈ E

and therefore T

m

(A) ∈ L

N

(

^m

E

).

^QED

Next we will see that the operators from the Nicodemi sequence preserve multilinear mappings of ﬁnite type and nuclear multilinear mappings. We consider ﬁrst the case of scalar-valued multilinear mappings.

23 Theorem. Let R

m

: L(

^m

E) −→ L(

^m

F ) be a Nicodemi sequence. Then R

m

A ∈ L

Θ

(

^m

F ) for all A ∈ L

Θ

(

^m

E), where Θ = f or N .

Proof. We will write the proof in detail in the case Θ = N . In the case Θ = f , the proof is similar. Given A ∈ L

N

(

^m

E), there exist sequences (ϕ

ji

)

_i∈N

in E

, 1 ≤ j ≤ m, and (c

i

)

_i∈N

in K, with

_∞

i=1

ϕ

1i

· · · ϕ

mi

|c

i

| < ∞ such that A(x

₁

, . . . , x

m

) =

_∞

i=1

ϕ

_1i

(x

₁

) · · · ϕ

mi

(x

m

)c

i

for all (x

₁

, . . . , x

m

) ∈ E

^m

. Let T

m

: L(

^m

E) −→ L(

^m

E

) be the Nicodemi sequence beginning with the natural embedding T

₁

= J

E

: E

→ E

. By Theorem 9 we have that

R

m

A(y

₁

, . . . , y

m

) = T

m

A(R

₁

(J

F

y

₁

), . . . , R

₁

(J

F

y

m

))

for all A ∈ L(

^m

E), and y

₁

, . . . , y

m

∈ F , where R

₁

is the transpose of R

₁

and by Lemma 22 we have that

T

m

(A)(x

₁

, . . . , x

_m

) =

∞ i=1

(T

₁

ϕ

_1i

)(x

₁

) · · · (T

1

ϕ

mi

)(x

_m

)c

i

for all (x

₁

, . . . , x

_m

) ∈ (E

)

^m

. Therefore

(14)

R

m

A(y

₁

, . . . , y

m

) =T

m

A(R

₁

(J

_F

y

₁

), . . . , R

₁

(J

_F

y

m

))

=

∞ i=1

(T

₁

ϕ

_1i

)(R

₁

J

F

y

₁

) · · · (T

1

ϕ

mi

)(R

₁

J

F

y

m

)c

i

=

∞ i=1

R

₁

ϕ

_1i

(y

₁

) · · · R

1

ϕ

mi

(y

m

)c

i

for all y

₁

, . . . , y

m

∈ F because < T

1

ϕ, R

₁

(J

F

y) >=< J

_E

ϕ, R

₁

(J

F

y) >=

< R

₁

(J

F

y), ϕ >=< J

F

y, R

₁

ϕ >=< R

₁

ϕ, y >, for all ϕ ∈ E

and all y ∈ F . Therefore

R

m

A =

∞ i=1

R

₁

ϕ

_1i

⊗ · · · ⊗ R

1

ϕ

mi

⊗ c

i

(15)

and then R

m

A ∈ L

N

(

^m

F ).

^QED

The next theorem considers the case of vector-valued multilinear mappings.

24 Theorem. Let R

_m

: L(

^m

E) −→ L(

^m

F ) be a Nicodemi sequence and let ( R

m

: L(

^m

E; G

) −→ L(

^m

F ; G

) be the corresponding sequence for vector-valued multilinear mappings. Then ( R

m

A ∈ L

Θ

(

^m

F ; G

) for all A ∈ L

Θ

(

^m

E; G

) where Θ = f or N .

Proof. We will only write the proof in the case Θ = N . In the case Θ = f , the proof is similar. Given A ∈ L

N

(

^m

E; G

) there exist sequences (ϕ

_ji

)

_i∈N

in E

, 1 ≤ j ≤ m, and (c

i

)

_i∈N

in G

, with

_∞

i=1

ϕ

1i

· · · ϕ

mi

c

i

< ∞ such that A =

_∞

i=1

ϕ

_1i

⊗ · · · ⊗ ϕ

mi

⊗ c

i

. Then δ

_z

◦ A =

_∞

i=1

ϕ

_1i

⊗ · · · ⊗ ϕ

mi

⊗ c

i

(z) for all z ∈ G, and clearly δ

z

◦ A ∈ L

N

(

^m

E). We have by (15) that

R

m

(δ

z

◦ A) =

∞ i=1

R

₁

ϕ

_1i

⊗ · · · ⊗ R

1

ϕ

mi

⊗ c

i

(z)

for all z ∈ G. Then

R (

m

A(y

₁

, . . . , y

m

)(z) =R

m

(δ

z

◦ A)(y

1

, . . . , y

m

)

=

∞ i=1

R

₁

ϕ

_1i

(y

₁

) · · · R

1

ϕ

mi

(y

m

)c

i

(z)

for all z ∈ G and y

1

, . . . , y

m

∈ F. We conclude that ( R

m

A =

∞ i=1

R

₁

ϕ

_1i

⊗ · · · ⊗ R

1

ϕ

mi

⊗ c

i

(16)

and then ( R

m

A ∈ L

N

(

^m

F ; G

).

^QED

(15)

25 Deﬁnition. Given a Nicodemi sequence R

m

: L(

^m

E; G) −→ L(

^m

F ; G), we deﬁne

R '

m

: P(

^m

E; G) −→ P(

^m

F ; G)

by ' R

m

A = ' R

m

A por every symmetric A ∈ L(

^m

E; G).

26 Lemma. Let T

m

: L(

^m

E) −→ L(

^m

E

) be the Nicodemi sequence beginning with the natural embedding T

₁

= J

E

: E

→ E

. Let P ∈ P

N

(

^m

E) and let c

₁

, . . . , c

n

∈ K, ϕ

1

, . . . , ϕ

m

∈ E

such that

P =

∞ i=1

ϕ

^m_i

⊗ c

i

.

Then

T '

m

(P ) =

∞ i=1

(T

₁

ϕ

i

)

^m

⊗ c

i

.

In particular ' T

m

(P ) ∈ P

N

(

^m

E

).

The proof of Lemma 26 is similar to the proof of Lemma 22 and is omit- ted. Next we will see that the operators from the Nicodemi sequence preserve homogeneous polynomials of ﬁnite type and nuclear homogeneous polynomials.

We consider ﬁrst the case of scalar-valued homogeneous polynomials.

27 Theorem. Let R

m

: L(

^m

E) −→ L(

^m

F ) be a Nicodemi sequence. Then R '

m

P ∈ P

Θ

(

^m

F ) for all P ∈ P

Θ

(

^m

E) where Θ = f or N .

Proof. We will only write the proof in the case Θ = N . In the case Θ = f , the proof is similar. Given P ∈ P

N

(

^m

E), there exist (c

_i

)

_i∈N

∈ K, (ϕ

i

)

_i∈N

∈ E

such that P can be written in the form: P (x) =

_∞

i=1

ϕ

_i

(x)

^m

c

_i

for all x ∈ E. Let T

_m

: L(

^m

E) −→ L(

^m

E

) be the Nicodemi sequence beginning with the natural embedding T

₁

= J

_E

: E

→ E

. By Theorem 9 we have that

R

_m

A(y

₁

, . . . , y

_m

) = T

_m

A(R

₁

(J

_F

y

₁

), . . . , R

₁

(J

_F

y

_m

))

for all A ∈ L(

^m

E), and y

₁

, . . . , y

m

∈ F , where R

₁

is the transpose of R

₁

and by Lemma 26 we have that ' T

_m

(P ) has the following form:

T '

m

(P )(x

) =

∞ i=1

(T

₁

ϕ

i

)(x

)

^m

c

i

for all x

∈ E

. Therefore, A being the m-linear mapping associated with P , it

(16)

follows that

R '

m

P (y) =R

m

A(y, . . . , y ) *+ ,

m−times

)

=T

m

A(R

₁

(J

F

y), . . . , R

₁

(J

F

y)

) *+ ,

m−times

)

= ' T

_m

P (R

₁

(J

_F

y))

=

∞ i=1

(T

₁

ϕ

_i

)(R

₁

(J

_F

y))

^m

c

_i

=

∞ i=1

(R

₁

ϕ

i

y))

^m

c

i

for all y ∈ F . Therefore,

R '

_m

P =

∞ i=1

(R

₁

ϕ

_i

)

^m

⊗ c

i

(17)

and then ' R

_m

P ∈ P

N

(

^m

F ).

^QED

The next theorem consider the case of vector-valued homogeneous polynomials.

28 Theorem. Let R

m

: L(

^m

E) −→ L(

^m

F ) be a Nicodemi sequence, let ( R

m

: L(

^m

E; G

) −→ L(

^m

F ; G

) be the corresponding Nicodemi sequence for vector- valued multilinear mappings. Then ' R

m

P ∈ P

Θ

(

^m

F ; G

) for all P ∈ P

Θ

(

^m

E; G

) where Θ = f or N .

Proof. We will only write the proof in the case Θ = N . In the case Θ = f , the proof is similar. Given ' A ∈ P

N

(

^m

E; G

), there exist sequences (ϕ

i

)

_i∈N

in E

, and (c

i

)

_i∈N

in G

with

_∞

i=1

ϕ

i

^m

c

i

< ∞ such that ' A(x) =

_∞

i=1

ϕ

i

(x)

^m

c

i

for all x ∈ E, where A ∈ L

^s

(

^m

E; G

). We observe that if B =

_∞

i=1

ϕ )

i

⊗ · · · ⊗ ϕ *+ ,

i m−times

⊗c

i

, then B ∈ L

N

(

^m

E; G

) ∩ L

^s

(

^m

E; G

) and ' B = ' A. Then

A = B from the injectivity of the canonical isomorphism L

^s

(

^m

E; G

) −→

P(

^m

E; G

). Thus, we get that A ∈ L

N

(

^m

E; G

) and by (16) ( R

_m

A =

∞ i=1

R

₁

ϕ

_i

⊗ · · · ⊗ R

1

ϕ

_i

) *+ ,

m−times

⊗c

i

.

Since ' R

_m

A = ' ( R

_m

A, it follows that ' R

_m

P =

n i=1

(R

₁

ϕ

i

)

^m

⊗ c

i

(18)

(17)

and then ' R

m

P ∈ P

N

(

^m

F ; G

).

^QED

We remark that if E is a closed subspace of F , then Aron and Berner [1, Theorem 2.1] proved that the restriction mapping

P

N

(

^m

F ; G) −→ P

N

(

^m

E; G)

is surjective for each m ∈ N, but even in the case of the natural embedding J

_E

: E → E

, they did not study the problem of existence of a linear extension operator

T

_m

: P

N

(

^m

E) −→ P

N

(

^m

E

)

for each m ∈ N. Next we will see that each isomorphism between E

and F

induces an isomorphism between L

_Θ

(

^m

E; G

) and L

_Θ

(

^m

F ; G

) for each m ∈ N and induces also an isomorphism between P

Θ

(

^m

E; G

) and P

Θ

(

^m

F ; G

) for each m ∈ N, where Θ = f or N.

29 Theorem. If E

and F

are isomorphic, then L

_Θ

(

^m

E) and L

_Θ

(

^m

F ) are isomorphic, for all m ∈ N, where Θ = f or N.

Proof. We will only write the proof in the case Θ = N . In the case Θ = f , the proof is similar. We use the notations from the proof of Theorem 17 By Theorem 23 we have that R

_m

(L

_N

(

^m

E)) ⊂ L

N

(

^m

F ). As in the proof of Theorem 23 we can prove that S

_m

(R

_m

A) =

_∞

i=1

S

₁

(R

₁

ϕ

_1i

) ⊗ · · · ⊗ S

1

(R

₁

ϕ

_mi

) ⊗ c

i

. Since S

₁

◦ R

1

is the identity, we have that S

_m

(R

_m

A) =

_∞

i=1

ϕ

_1i

⊗ · · · ⊗ ϕ

mi

⊗ c

i

= A for all A ∈ L

N

(

^m

E). On the other hand, in a similar way, we can get that S

_m

(L

_N

(

^m

F )) ⊂ L

N

(

^m

E) and R

_m

(S

_m

B) = B for all B ∈ L

N

(

^m

F ). We conclude that L

_N

(

^m

E) and L

_N

(

^m

F ) are isomorphic for all m ∈ N.

^QED

30 Theorem. If E

and F

are isomorphic, then L

_Θ

(

^m

E; G

) and L

_Θ

(

^m

F ; G

) are isomorphic for all m ∈ N, where Θ = f or N.

Proof. We will only write the proof in the case Θ = N . In the case Θ = f , the proof is similar. We use the notations from the proof of Theo- rem 19. By Theorem 24 we have that ( R

m

(L

N

(

^m

E; G

)) ⊂ L

N

(

^m

F ; G

) and S

_m

(L

_N

(

^m

F ; G

)) ⊂ L

N

(

^m

E; G

). As in the proof of Theorem 24 we can prove that S

m

(( R

m

A) =

_∞

i=1

S

₁

(R

₁

ϕ

_1i

) ⊗ · · · ⊗ S

1

(R

₁

ϕ

mi

) ⊗ c

i

. Since S

₁

◦ R

1

is the identity, we have that S

_m

(( R

_m

A) =

_∞

i=1

ϕ

_1i

⊗ · · · ⊗ ϕ

mi

⊗ c

i

= A for

all A ∈ L

N

(

^m

E; G

). On the other hand, in a similar way, we can get that

R (

m

( S

m

B) = B for all B ∈ L

N

(

^m

F ; G

). We conclude that L

N

(

^m

E; G

) and

L

N

(

^m

F ; G

) are isomorphic for all m ∈ N.

^QED

31 Theorem. If E

and F

are isomorphic, then P

Θ

(

^m

E) and P

Θ

(

^m

F ) are

isomorphic for all m ∈ N, where Θ = f or N.

(18)

Proof. We will only write the proof in the case Θ = N . In the case Θ = f , the proof is similar. We use the notations from the proof of Theorem 17.

By Theorem 27 we have that ' R

m

(P

N

(

^m

E)) ⊂ P

N

(

^m

F ) and ' S

m

(P

N

(

^m

F )) ⊂ P

N

(

^m

E). As in the proof of Theorem 27 we can prove that ' S

m

( ' R

m

P ) =

_∞

i=1

(S

₁

(R

₁

ϕ

i

))

^m

⊗ c

i

. Since S

₁

◦ R

1

is the identity, we have that ' S

m

( ' R

m

P ) =

_∞

i=1

(ϕ

i

)

^m

⊗ c

i

= P for all P ∈ P

N

(

^m

E). On the other hand, in a similar way, we can get that ' R

m

( ' S

m

Q) = Q for all Q ∈ P

N

(

^m

F ). We conclude that P

N

(

^m

E) and P

N

(

^m

F ) are isomorphic for all m ∈ N.

^QED

32 Theorem. If E

and F

are isomorphic, then P

Θ

(

^m

E; G

) and P

Θ

(

^m

F ; G

) are isomorphic for all m ∈ N, where Θ = f or N.

Proof. We will only write the proof in the case of Θ = N . In the case of Θ = f the proof is similar. We use the notations from the proof of The- orem 19. By Theorem 28 we have that ' R

_m

( P

N

(

^m

E; G

)) ⊂ P

N

(

^m

F ; G

) and ' S

m

( P

N

(

^m

F ; G

)) ⊂ P

N

(

^m

E; G

). As in the proof of Theorem 28 we can prove that ' S

m

( ' R

m

P ) =

_∞

i=1

(S

₁

(R

₁

ϕ

i

))

^m

⊗ c

i

. Since S

₁

◦ R

1

is the identity, we have that ' S

m

( ' R

m

P ) =

_∞

i=1

(ϕ

i

)

^m

⊗ c

i

= P for all P ∈ P

N

(

^m

E; G

). On the other hand, in a similar way, we can get that ' R

_m

(' S

_m

Q) = Q for all Q ∈ P

N

(

^m

F ; G

). We conclude that P

N

(

^m

E; G

) and P

N

(

^m

F ; G

) are isomor-

phic for all m ∈ N.

^QED

Let us notice that in Theorems 29, 30, 31 and 32 we do not need the Arens regularity hypothesis.

3 Isomorphisms between spaces of compact or weakly compact multilinear mappings or homogeneous polynomials.

We ﬁrst show that the operators ( R

m

from the Nicodemi sequence preserve compact and weakly compact multilinear mappings.

33 Theorem. Let R

_m

: L(

^m

E) −→ L(

^m

F ) be a Nicodemi sequence and let ( R

m

: L(

^m

E; G

) −→ L(

^m

F ; G

) be the corresponding Nicodemi sequence for vector-valued multilinear mappings. Then ( R

Isomorphisms between spaces of multilinear mappings or homogeneous polynomials