What is “local theory of Banach spaces”?

What is “local theory of Banach spaces”?

by

A L B R E C H T P I E T S C H

^∗

(Jena)

Abstract. Banach space theory splits into several subtheories. On the one hand, there are an isometric and an isomorphic part; on the other hand, we speak of global and local aspects. While the concepts of isometry and isomorphy are clear, everybody seems to have its own interpretation of what “local theory” means. In this essay we analyze this situation and propose rigorous definitions, which are based on new concepts of local representability of operators.

Preamble. Of course, the quality of a theorem does not depend on the fact to which theory it belongs. Nevertheless, in order to systematize our knowledge we need criteria that enable us to collect similar results in theories.

1. Historical roots

1.1. Let us begin with some quotations.

Pe lczy´ nski and Rosenthal [p–r], p. 263: Localization refers to obtaining quantitative finite-dimensional formulations of infinite-dimensional results.

Tomczak-Jaegermann [TOM], p. 5: A property (of Banach spaces or of operators acting between them) is called local if it can be defined by a quantitative statement or inequality concerning a finite number of vectors or finite-dimensional subspaces.

Lindenstrauss and Milman [l–m], p. 1151: The name ‘local theory’ is applied to two somewhat different topics:

(a) The quantitative study of n-dimensional normed spaces as n → ∞.

(b) The relation of the structure of an infinite-dimensional space and its finite-dimensional subspaces.

1991 Mathematics Subject Classification: 46B07, 46B08.

∗

The proof of the Basic Lemma in 5.9 is due to W. B. Johnson.

Research supported by DFG grant PI 322/1-2.

[273]

1.2. In 1956, Grothendieck defined the following concept:

L’espace norm´ e E a un type lin´eaire inf´erieur ` a celui d’un espace norm´ e F , si on peut trouver un M > 0 fixe tel que tout sous-espace de dimension finie E

1

de E soit isomorphe “` a M pr`es” ` a un sous-espace de dimension finie F

1

de F (Banach–Mazur distance d(E

1

, F

1

) ≤ 1 + M); et que E a un type m´ etrique inf´ erieur ` a celui de F , si la condition pr´ec´edente est satisfaite pour tout M > 0.

In his theory of super-reflexivity, James used to speak of crudely finite representability and of finite representability, respectively.

1.3. A decisive step was done when Dacunha-Castelle and Krivine intro- duced the technique of ultraproducts in Banach space theory. It soon turned out that X is finitely representable in Y if and only if X is isometric to a subspace of some ultrapower Y

^U

.

Ultraproducts of operators were studied by Pietsch. Next, Beauzamy introduced two (possibly different) concepts of finite representability for operators. A more appropriate approach is due to Heinrich.

1.4. Following Brunel and Sucheston, a property P is called a super- property if it carries over from Y to all X finitely representable in Y . Equiv- alently, this means that P is preserved under the formation of ultrapowers and subspaces.

2. Notation

2.1. Let L stand for the class of all (real or complex) Banach spaces.

Denote the set of all (bounded linear) operators from X into Y by L(X, Y ), and write L := S L(X, Y ), where the union ranges over X, Y ∈ L.

2.2. An operator J ∈ L(X, Y ) is an injection if there exists a constant c > 0 such that kJxk ≥ ckxk for all x ∈ X. In the case when kJxk = kxk, we speak of a metric injection. For every (closed linear) subspace M of X, the canonical embedding from M into X is denoted by J

_M^X

.

2.3. An example of a metric injection is the map K

X

: X → X

^∗∗

that assigns to every x ∈ X the functional x

^∗

7→ hx, x

^∗

i. For T ∈ L(X, Y ), we let T

^reg

:= K

Y

T , where the superscript

^reg

stands for regular .

2.4. Let I be an index set. We denote by l

∞

(I) the Banach space of bounded scalar families (ξ

i

) with the norm k(ξ

ⁱ

) | l

^∞

(I) k := sup

_i∈I

|ξ

ⁱ

|.

Note that l

∞

(I) has the metric extension property. This means that every operator T ∈ L(M, l

^∞

(I)) defined on a subspace M of X admits a norm- preserving extension T

^ext

∈ L(X, l

^∞

(I)).

If x ∈ X, then (hx, x

^∗

i) can be viewed as an element of l

^∞

(B

X^∗

), where

B

X^∗

is the closed unit ball of X

^∗

. In this way, we get the canonical injection

J

_X^inj

from X into X

^inj

:= l

∞

(B

X^∗

). For T ∈ L(X, Y ), we let T

^inj

:= J

_Y^inj

T , where the superscript

^inj

stands for injective.

2.5. An operator Q from X onto Y is called a surjection. In the case when the open unit ball of X is mapped onto the open unit ball of Y , we speak of a metric surjection. For every (closed linear) subspace N of Y , the quotient map from Y onto Y /N is denoted by Q

^Y_N

.

2.6. Given Banach spaces X

i

with i ∈ I, we denote by [l

^∞

(I), X

i

] the Banach space of all bounded families (x

_i

) such that x

_i

∈ X

i

. Fix an ultra- filter U on the index set I. The collection of all equivalence classes

(x

i

)

^U

:= {(x

^◦i

) ∈ [l

^∞

(I), X

i

] : U-lim

_i

kx

^◦i

− x

ⁱ

k = 0}

is a Banach space (X

i

)

^U

under the norm k(x

ⁱ

)

^U

k := U-lim

ⁱ

kx

ⁱ

k.

Next, let T

_i

∈ L(X

i

, Y

_i

) with i ∈ I be a bounded family of operators.

Then (T

i

)

^U

: (x

i

)

^U

7→ (T

ⁱ

x

i

)

^U

defines an operator from (X

i

)

^U

into (Y

i

)

^U

such that k(T

i

)

^U

k = U-lim

i

kT

i

k.

These new objects are referred to as ultraproducts.

2.7. If X

i

= X, Y

i

= Y and T

i

= T , then we speak of ultrapowers, de- noted by X

^U

, Y

^U

and T

^U

, respectively. In this case, there exists a canonical map J

_X^U

from X into X

^U

, which sends x to (x

i

) with x

i

= x. Moreover, we define Q

^U_X

: (x

_i

)

^U

7→ U-lim

i

K

_X

x

_i

. The right-hand limit is taken with respect to the weak

^∗

topology of X

^∗∗

. These constructions yield the formula T

^reg

= K

Y

T = Q

^U_Y

T

^U

J

_X^U

for T ∈ L(X, Y ). Unfortunately, the canonical embedding K

Y

cannot be avoided.

2.8. Throughout this essay, we let

T ∈ L(X, Y ), T

0

∈ L(X

0

, Y

0

), T

1

∈ L(X

1

, Y

1

), etc.

3. Subtheories. Imitating Felix Klein, we now describe an Erlanger Programm for operators in Banach spaces.

3.1. Suppose we are given a preordering ≺ on L, the class of all opera- tors. That is, T

0

≺ T

1

and T

1

≺ T

2

imply T

0

≺ T

2

, and we have T ≺ T . A property is said to be ≺-stable if it is inherited from T

¹

to every T

0

≺ T

¹

. So every preordering ≺ leads to a subtheory that deals with the associated stable properties.

Writing T

0

∼ T

1

whenever T

0

≺ T

1

and T

1

≺ T

0

yields an equivalence relation. Of course, we could also study the concept of ∼-stability.

3.2. Note that a preordering on L, the class of all Banach spaces, is

obtained by letting X

0

≺ X

1

if I

X0

≺ I

X1

, where I

X0

and I

X1

denote

the identity maps of X

0

∈ L and X

¹

∈ L, respectively. In this way, every

subtheory of operators contains a subtheory of spaces, as a special part.

4. Global representability

4.1. An operator T

0

∈ L(X

⁰

, Y

0

) is globally representable in an operator T

1

∈ L(X

¹

, Y

1

) if there exist A

1

∈ L(X

⁰

, X

1

) and B

1

∈ L(Y

¹

, Y

0

) such that T

0

= B

1

T

1

A

1

. That is,

X

1

Y

1

X

0

Y

0

T1

//

B1



A1

OO

T0

//

In this case, we write T

0

glo

≺ T

1

. Of course,

^glo

≺ is a preordering on L. The corresponding stable properties are said to be global. Obviously, X

0

glo

≺ X

1

means that X

₀

is isomorphic to a complemented subspace of X

₁

.

4.2. A property is called injective if it carries over from T

^inj

∈ L(X, Y

^inj

) to T ∈ L(X, Y ).

4.3. The preordering T

0 injglo

≺ T

¹

is defined by T

₀^{inj glo}

≺ T

1^inj

. Note that T

₀ ^glo

≺ T

1

implies T

₀ ^injglo

≺ T

1

. A property turns out to be

^injglo

≺ -stable if and only if it is simultaneously injective and global, which justifies the symbol

injglo

≺ . Clearly, X

0 injglo

≺ X

1

means that X

₀

is isomorphic to a subspace of X

1

.

4.4. So far, we have discussed isomorphic notions. Taking into account isometric aspects, we could define a global preordering by assuming that the operators A

₁

and B

₁

in T

₀

= B

₁

T

₁

A

₁

satisfy the condition kB

1

k · kA

1

k ≤ 1.

We may also require that, for any choice of ε > 0, there exists a factorization such that kB

1

k · kA

1

k ≤ 1 + ε. Another possibility would be to assume that A

₁

is an injection and that B

₁

is a surjection, metric or not.

5. Operators between finite-dimensional spaces

5.1. The symbol F stands for the collection of all finite-dimensional Banach spaces. Note that, upon identifying isometric copies, F is a set.

Throughout, let E ∈ F and F ∈ F. The underlying (real or complex) scalar field (sometimes viewed as a 1-dimensional Banach space) is denoted by K.

5.2. With every operator T ∈ L(X, Y ) we associate the germs

L(T | E, F ) := {BT A ∈ L(E, F ) : kA : E → Xk ≤ 1, kB : Y → F k ≤ 1}.

This definition is illustrated by the following diagram:

X Y

E F

T

//

B



A

OO

BT A

//

Roughly speaking, local properties of T can be formulated in terms of the family {L(T | E, F ) : E, F ∈ F}. Working with the closed hulls L(T | E, F ) will turn out to be more elegant.

5.3. The sets L(T | E, F ) are bounded and circled, but may look like a hedgehog. For example, L(Id : K → K | E, F ) consists of all S ∈ L(E, F ) with rank(S) ≤ 1 and kSk ≤ 1. In the case when T does not attain its norm, L(T | K, K) is an open disc. On the other hand, L(T | E, F ) may also be very nice as shown by L(Id : l

²

→ l

²

| E, F ), which is the closed unit ball of a norm.

5.4. Our first result is obvious.

Proposition. L(T | E, F ) = L(T

^reg

| E, F ).

5.5. We now state one of the most important tools of local theory.

Principle of Local Reflexivity. Let M and N be finite-dimensional subspaces of X

^∗∗

and X

^∗

, respectively. Then for every ε > 0 there exists an isomorphism I

_ε

from M onto some subspace M

_ε

of X such that kI

ε

k ≤ 1+ε, kI

ε⁻¹

k ≤ 1 + ε and

hx

^∗∗

, x

^∗

i = hI

^ε

x

^∗∗

, x

^∗

i for all x

^∗∗

∈ M and x

^∗

∈ N.

5.6. The correspondence S 7→ S

^∗

defines an isometry between L(E, F ) and L(F

^∗

, E

^∗

). The image of a set L will be denoted by L

^∗

.

Proposition.

L

^∗

(T | E, F ) ⊆ L(T

^∗

| F

^∗

, E

^∗

) ⊆ (1 + ε)L

^∗

(T | E, F ) for all ε > 0.

P r o o f. The left-hand inclusion is trivial.

By definition, every operator S

^∗

∈ L(T

^∗

| F

^∗

, E

^∗

) can be decomposed in the form S

^∗

= U T

^∗

V , where kV : F

^∗

→ Y

^∗

k ≤ 1 and kU : X

^∗

→ E

^∗

k ≤ 1.

Putting B := K

_F⁻¹

V

^∗

K

Y

, we get V = B

^∗

. The principle of local reflexivity provides us with an isomorphism I

ε

from U

^∗

K

E

(E) onto some subspace M

ε

of X such that kI

ε

k ≤ 1 + ε and

hU

^∗

K

_E

e, T

^∗

V f

^∗

i = hI

ε

U

^∗

K

_E

e, T

^∗

V f

^∗

i for e ∈ E and f

^∗

∈ F

^∗

. Letting A := I

ε

U

^∗

K

E

, we obtain kA : E → Xk ≤ 1 + ε and

hBT Ae, f

^∗

i = hI

ε

U

^∗

K

E

e, T

^∗

V f

^∗

i = hU

^∗

K

E

e, T

^∗

V f

^∗

i = he, S

^∗

f

^∗

i

for e ∈ E and f

^∗

∈ F

^∗

. So S = BT A ∈ (1 + ε)L(T | E, F ).

5.7. We now formulate a corollary of the preceding result.

Proposition.

L(T | E, F ) ⊆ L(T

^∗∗

| E, F ) ⊆ (1 + ε)L(T | E, F ) for all ε > 0.

5.8. The next statement even holds for compact operators. However, we will only need the finite-dimensional case.

Proposition. If T has finite rank , then L(T

^∗

| F

^∗

, E

^∗

) is closed.

P r o o f. The closed unit balls

{U : kU : X

^∗

→ E

^∗

k ≤ 1} and {V : kV : F

^∗

→ Y

^∗

k ≤ 1}

are compact in the weak

^∗

topologies induced by the seminorms p

_e,x∗

(U ) := |he, Ux

^∗

i| and p

y,f^∗

(V ) := |hy, V f

^∗

i|.

Moreover, the norm topology of L(F

^∗

, E

^∗

) coincides with the weak

^∗

topology obtained from p

e,f^∗

(S

^∗

) := |he, S

^∗

f

^∗

i|. Since T can be written in the form

T =

N

X

i=1

x

^∗_i

⊗ y

i

, we get

he, UT

^∗

V f

^∗

i =

N

X

i=1

he, Ux

^∗i

ihy

i

, V f

^∗

i.

So the bilinear map (U, V ) 7→ UT

^∗

V is continuous, which in turn implies the compactness of L(T

^∗

| F

^∗

, E

^∗

).

5.9. I am very indebted to W. B. Johnson for a proof of the following result, which is the crucial device in our understanding of local theory.

Basic Lemma. L(T | E, F ) ⊆ (1 + ε)L(T | E, F ) for all ε > 0.

P r o o f. If T has finite rank, then it follows from the previous proposition that L(T

^∗∗

| E, F ) is closed. Hence, by 5.7,

L(T | E, F ) ⊆ L(T

^∗∗

| E, F ) ⊆ (1 + ε)L(T | E, F ).

To treat the case rank(T ) = ∞, we let m := dim(E) and n := dim(F ).

If d := 2m + n, then there exist A

0

∈ L(l

2^d

, X) and B

0

∈ L(Y, l

2^d

) such that

B

₀

T A

₀

is the identity map of l

^d₂

. Given S ∈ L(T | E, F ) and δ > 0, we find a

decomposition S = S

1

+ S

2

such that S

1

∈ L(T | E, F ) and kS

²

k ≤ δ. Write

S

1

= B

1

T A

1

with kA

1

k ≤ 1 and kB

1

k ≤ 1. Let M be the range of B

0

T A

1

,

and let N be the null space of B

1

T A

0

. Then cod(M

^⊥

) = dim(M ) ≤ m

and cod(N ) ≤ n. Hence cod(M

^⊥

∩ N) ≤ m + n, which is equivalent to

dim(M

^⊥

∩ N) ≥ m. This implies that M

^⊥

∩ N contains an m-dimensional

subspace H. Let J denote the embedding from H into l

₂^d

, while Q stands

for the orthogonal projection from l

^d₂

onto H. So QB

0

T A

0

J is the identity

map of H. John’s theorem provides us with an isomorphism U ∈ L(E, H) such that kUk · kU

⁻¹

k ≤ √

m.

E F

X Y

N ⊂ l

2^d

l

^d₂

⊃ M

H H

E E

S1

//

A1



_T

//

B0



B1

OO

A0

OO

Id

//

Q



J

OO

IH

//

U⁻¹



U

OO

IE

//

We obtain S

₂

= S

₂

U

⁻¹

QB

₀

T A

₀

JU = B

₂

T A

₂

, where B

₂

:= S

₂

U

⁻¹

QB

₀

and A

2

:= A

0

JU . The main purpose of the construction above was to get the formulas B

1

T A

0

J = O and QB

0

T A

1

= O, which in turn yield

B

1

T A

2

= B

1

T A

0

JU = O and B

2

T A

1

= S

2

U

⁻¹

QB

0

T A

1

= O.

Hence

S = S

1

+ S

2

= B

1

T A

1

+ B

2

T A

2

= (B

1

+ B

2

)T (A

1

+ A

2

).

Moreover,

kB

2

k · kA

2

k ≤ kS

2

U

⁻¹

QB

₀

k · kA

0

JU k ≤ δ √

m kB

0

k · kA

0

k.

Clearly, U may be chosen such that kA

2

k = kB

2

k. Then

kA

¹

+ A

2

k · kB

¹

+ B

2

k ≤ (1 + pδ√m kB

⁰

k · kA

⁰

k )

²

≤ 1 + ε

whenever δ > 0 is sufficiently small. This proves that S ∈ (1+ε)L(T | E, F ).

5.10. The preceding result can be stated in the following form.

Theorem. L(T | E, F ) = T

ε>0

(1 + ε) L(T | E, F ).

5.11. In the theory of ultraproducts, we have a striking counterpart of the principle of local reflexivity; see [kue], [ste] and [hei 1], p. 8.

K¨ ursten–Stern Lemma. Let M and N be finite-dimensional sub- spaces of (X

i

)

^U

and ((X

i

)

^U

)

^∗

, respectively. Then for every ε > 0 there exists an isomorphism I

_ε

from N onto some subspace N

_ε

of (X

_i^∗

)

^U

such that kI

^ε

k ≤ 1 + ε, kI

ε⁻¹

k ≤ 1 + ε and

hx, x

^∗

i = hx, I

ε

x

^∗

i for all x ∈ M and x

^∗

∈ N.

5.12. For completeness, we provide another standard result of the theory of ultraproducts.

Lemma. Assume that x

₁

= (x

1i

)

^U

, . . . , x

m

= (x

mi

)

^U

∈ (X

i

)

^U

are lin- early independent. Then, given ε > 0, there exists U ∈ U such that

m

X

h=1

ξ

h

x

hi

≤ (1 + ε)

m

X

h=1

ξ

h

x

h

for ξ

1

, . . . , ξ

m

∈ K and i ∈ U.

P r o o f. First of all, fix δ > 0 and choose a finite δ-net N in the unit sphere S

^m₁

of l

^m₁

. Next, pick U ∈ U such that

  

m

X

h=1

ξ

h

x

hi

−

m

X

h=1

ξ

h

x

h

   ≤ δ

for (ξ

h

) ∈ N and i ∈ U. Clearly, we may assume that N contains the unit vectors of l

^m₁

. Then it follows that |kx

hi

k − kx

h

k| ≤ δ for h = 1, . . . , m and i ∈ U. Putting c := max{kx

¹

k, . . . , kx

^m

k}, we get

  

m

X

h=1

ξ

h

x

hi

−

m

X

h=1

ξ

h

x

_h

 

 ≤ (1 + 2c + δ)δ for (ξ

h

) ∈ S

^m1

and i ∈ U. Hence, by homogeneity,

m

X

h=1

ξ

h

x

hi

≤

m

X

h=1

ξ

h

x

h

+ (1 + 2c + δ)δ

m

X

h=1

|ξ

^h

|

for ξ

1

, . . . , ξ

m

∈ K and i ∈ U. Since x

¹

, . . . , x

m

are linearly independent, we find a constant b > 0 such that

m

X

h=1

|ξ

h

| ≤ b

m

X

h=1

ξ

h

x

h

for ξ

1

, . . . , ξ

m

∈ K, which implies

m

X

h=1

ξ

h

x

hi

≤ (1 + (1 + 2c + δ)bδ)

m

X

h=1

ξ

h

x

_h

for ξ

1

, . . . , ξ

m

∈ K and i ∈ U. Choosing δ > 0 sufficiently small completes the proof.

5.13. We are now in a position to establish an analogue of 5.7.

Proposition.

L(T | E, F ) ⊆ L(T

^U

| E, F ) ⊆ (1 + ε)L(T | E, F ) for all ε > 0.

P r o o f. The left-hand inclusion follows from T

^reg

= Q

^U_Y

T

^U

J

_X^U

; see 2.7.

Decompose the operator S ∈ L(T

^U

| E, F ) in the form S = BT

^U

A with kA : E → X

^U

k ≤ 1 and kB : Y

^U

→ F k ≤ 1. Write

A =

m

X

h=1

e

^∗_h

⊗ x

h

and B =

n

X

k=1

y

_k^∗

⊗ f

k

,

where x

1

, . . . , x

m

∈ X

^U

and y

₁^∗

, . . . , y

_n^∗

∈ (Y

^U

)

^∗

are linearly independent.

By 5.11, we find an isomorphism I

ε

from F := span {y

^∗1

, . . . , y

^∗_n

} onto some subspace F

ε

of (Y

^∗

)

^U

such that kI

^ε

k ≤ 1 + ε and

hT

^U

x

_h

, y

^∗_k

i = hT

^U

x

_h

, I

ε

y

^∗_k

i for h = 1, . . . , m and k = 1, . . . , n.

Fix representations x

h

= (x

hi

)

^U

and I

ε

y

_k^∗

= (y

_ki^∗

)

^U

with x

hi

∈ X and y

_ki^∗

∈ Y

^∗

. By the preceding lemma, there exists U ∈ U such that

m

X

h=1

ξ

h

x

hi

≤ (1 + ε)

m

X

h=1

ξ

h

x

h

and

n

X

k=1

η

_k

y

_ki^∗

≤ (1 + ε)

n

X

k=1

η

_k

y

^∗_k

for ξ

₁

, . . . , ξ

_m

∈ K, η

1

, . . . , η

_n

∈ K and i ∈ U. Letting

A

_i

:=

m

X

h=1

e

^∗_h

⊗ x

hi

and B

_i

:=

n

X

k=1

y

_ki^∗

⊗ f

k

yields operators with kA

i

: E → Xk ≤ 1 + ε and kB

i

: Y → F k ≤ 1 + ε.

Moreover, in view of

hT

^U

x

_h

, y

^∗_k

i = hT

^U

x

_h

, I

ε

y

_k^∗

i = U-lim

i

hT x

hi

, y

_ki^∗

i, it may be achieved that

kBT

^U

A − B

i

T A

_i

k =

m

X

h=1 n

X

k=1

( hT

^U

x

_h

, y

^∗_k

i − hT x

hi

, y

_ki^∗

i)e

^∗h

⊗ f

k

becomes as small as we please. So

S = BT

^U

A ∈ (1 + ε)

²

L(T | E, F ) ⊆ (1 + ε)

³

L(T | E, F ).

5.14. Finally, we summarize the most important results of this section.

Theorem. The closed germs L(· | E, F ) coincide for the operators T , T

^reg

, T

^∗∗

and T

^U

.

That is, from the local point of view we cannot distinguish between T , T

^reg

, T

^∗∗

and T

^U

.

6. Local representability

6.1. Let c ≥ 0. We say that an operator T

⁰

∈ L(X

⁰

, Y

0

) is locally

c-representable in an operator T

1

∈ L(X

1

, Y

1

) if for ε > 0, for any choice of

finite-dimensional spaces E and F , for A

0

∈ L(E, X

⁰

) and B

0

∈ L(Y

⁰

, F ) there exist A

1

∈ L(E, X

1

) and B

1

∈ L(Y

1

, F ) such that

B

1

T

1

A

1

= B

0

T

0

A

0

and kB

¹

k · kA

¹

k ≤ (c + ε)kB

⁰

k · kA

⁰

k.

In other words, we assume that

L(T

⁰

| E, F ) ⊆ (c + ε)L(T

¹

| E, F ) whenever ε > 0 or, by 5.10, that

L(T

0

| E, F ) ⊆ cL(T

1

| E, F ).

Roughly speaking, it is required that the finite-dimensional structure of T

1

is at least as rich as that of T

0

.

6.2. An operator T

0

∈ L(X

⁰

, Y

0

) is said to be locally representable in an operator T

1

∈ L(X

1

, Y

1

) if the above conditions hold for some c ≥ 0. In this case, we write T

0

loc

≺ T

¹

. Of course,

^loc

≺ is a preordering on L.

6.3. We now restate Theorem 5.14.

Theorem. The operators T , T

^reg

, T

^∗∗

and T

^U

are locally equivalent.

6.4. We see from 5.6 that the concept of local representability is stable under duality.

Proposition. Let c ≥ 0. Then the following are equivalent:

(1) T

₀

is locally c-representable in T

₁

. (2) T

₀^∗

is locally c-representable in T

₁^∗

.

6.5. The local nature of the above notion can also be seen from the next criterion.

Proposition. Let c ≥ 0. Then the following are equivalent:

(1) T

0

is locally c-representable in T

1

.

(2) For ε > 0, for x

01

, . . . , x

0m

∈ X

0

and for y

₀₁^∗

, . . . , y

^∗_0n

∈ Y

0^∗

we can find x

₁₁

, . . . , x

_1m

∈ X

1

and y

₁₁^∗

, . . . , y

_1n^∗

∈ Y

1^∗

such that

hT

1

x

_1h

, y

^∗_1k

i = hT

0

x

_0h

, y

^∗_0k

i for h = 1, . . . , m and k = 1, . . . , n,

m

X

h=1

ξ

h

x

1h

≤

m

X

h=1

ξ

h

x

0h

and

n

X

k=1

η

_k

y

^∗_1k

≤ (c + ε)

n

X

k=1

η

_k

y

_0k^∗

whenever ξ

1

, . . . , ξ

m

∈ K and η

¹

, . . . , η

n

∈ K.

P r o o f. We consider the operators A

0

:= J

_M^X0

and B

0

:= Q

^Y_N⁰

, where

M := span {x

01

, . . . , x

0m

} and N := {y ∈ Y

0

: hy, y

^∗01

i = . . . = hy, y

^∗0n

i = 0}.

Note that (Y

0

/N )

^∗

can be identified with span {y

^∗01

, . . . , y

_0n^∗

}. Choose A

¹

and B

1

such that

B

1

T

1

A

1

= B

0

T

0

A

0

, kA

1

k ≤ kA

0

k and kB

1

k ≤ (c + ε)kB

0

k.

Then we may put x

_1h

:= A

₁

x

_0h

and y

^∗_1k

:= B

₁^∗

x

^∗_0k

. This proves that (1) ⇒(2).

In order to verify the reverse implication, we represent A

0

∈ L(E, X

0

) and B

₀

∈ L(Y

0

, F ) in the form

A

₀

=

m

X

h=1

e

^∗_h

⊗ x

0h

and B

₀

=

n

X

k=1

y

^∗_0k

⊗ f

k

.

Pick x

11

, . . . , x

1m

and y

^∗₁₁

, . . . , y

_1n^∗

as described in (2). Then the operators A

1

:=

m

X

h=1

e

^∗_h

⊗ x

1h

and B

1

:=

n

X

k=1

y

_1k^∗

⊗ f

k

satisfy the conditions

B

1

T

1

A

1

= B

0

T

0

A

0

, kA

1

e k ≤ kA

0

e k and kB

1^∗

f

^∗

k ≤ (c + ε)kB

0^∗

f

^∗

k.

6.6. Next, we connect the concepts of local and global representability with the help of ultrapowers. This result should be compared with Theo- rem 1.2 in [hei 1].

Theorem. Let c ≥ 0. Then the following are equivalent:

(1) T

0

is locally c-representable in T

1

.

(2) There exist operators A ∈ L(X

0

, X

₁^U

) and B ∈ L(Y

1^U

, Y

₀^∗∗

), where U is an ultrafilter on a suitable index set I, such that

T

₀^reg

= BT

₁^U

A and kBk · kAk ≤ c.

P r o o f. In view of 5.13, we conclude from T

₀^reg

= BT

₁^U

A that L(T

⁰

| E, F ) = L(T

0^reg

| E, F ) ⊆ cL(T

1^U

| E, F ) ⊆ (1 + ε)cL(T

¹

| E, F ).

This proves that (2) ⇒(1).

We consider the index set I formed by all triples i = (M, N, ε). Here M is any finite-dimensional subspace of X

₀

, and N is any finite-codimensional subspace of Y

0

. As usual, ε > 0. For i = (M, N, ε) and i

0

= (M

0

, N

0

, ε

0

), we write i ≥ i

0

if M ⊇ M

0

, N ⊆ N

0

and 0 < ε ≤ ε

0

. Furthermore, fix some ultrafilter U on I that contains all sections {i ∈ I : i ≥ i

⁰

} with i

0

∈ I. Choose A

i

∈ L(M, X

1

) and B

i

∈ L(Y

1

, Y

0

/N ) such that kA

i

k ≤ 1, kB

i

k ≤ c + ε and B

i

T

₁

A

_i

= Q

^Y_N⁰

T

₀

J

_M^X0

. Let

x

i

:=  A

i

x if x ∈ M,

o if x 6∈ M.

Clearly, A : x 7→ (x

ⁱ

)

^U

defines an operator A ∈ L(X

⁰

, X

₁^U

) with kAk ≤ 1.

Next, given (y

i

)

^U

∈ Y

1^U

, we pick (y

^◦_i

)

^U

∈ Y

0^U

such that Q

^Y_N⁰

y

^◦_i

= B

i

y

i

and ky

i^◦

k ≤ (1 + ε)kB

i

y

_i

k. Put B : (y

i

)

^U

7→ U-lim

i

K

_Y0

y

_i^◦

, where the limit is taken with respect to the weak

^∗

topology of Y

₀^∗∗

. Then kBk ≤ c and T

₀^reg

= BT

₁^U

A.

6.7. A property is said to be

• local if it is

^loc

≺-stable,

• global if it is

^glo

≺ -stable,

• regular if it carries over from

T

^reg

∈ L(X, Y

^∗∗

) to T ∈ L(X, Y ),

• injective if it carries over from

T

^inj

∈ L(X, Y

^inj

) to T ∈ L(X, Y ),

• ultrapower-stable if, for all ultrafilters U, it is inherited from T ∈ L(X, Y ) to T

^U

∈ L(X

^U

, Y

^U

).

6.8. Thanks to 6.6, we are able to exhibit the typical ingredients of local representability.

Theorem. A property is local if and only if it is regular , ultrapower- stable and global.

7. Local operator schemes

7.1. We now define a preordering on the set of all operators acting between finite-dimensional spaces. If S

0

∈ L(E

⁰

, F

0

) and S

1

∈ L(E

¹

, F

1

), then S

0

≤ S

1

means that S

0

∈ L(S

1

| E

0

, F

0

). In other words, there exist A

1

and B

1

such that

S

0

= B

1

S

1

A

1

, kA

¹

: E

0

→ E

¹

k ≤ 1 and kB

¹

: F

1

→ F

⁰

k ≤ 1.

7.2. A local operator scheme is a rule that assigns to every pair (E, F ) of finite-dimensional Banach spaces a compact subset G(E, F ) of L(E, F ) such that the following conditions are satisfied:

(1) If S

0

∈ L(E

0

, F

0

) and S

1

∈ G(E

1

, F

1

), then S

0

≤ S

1

implies that S

₀

∈ G(E

0

, F

₀

).

(2) For S

1

∈ G(E

¹

, F

1

) and S

2

∈ G(E

²

, F

2

) there exists S ∈ G(E, F ) such that S

1

≤ S and S

²

≤ S.

(3) The sets G(E, F ) are uniformly bounded. That is, kSk ≤ c for all S ∈ G(E, F ), where the constant c > 0 does not depend on E and F .

7.3. The closed germs G(T | E, F ) := L(T | E, F ) of any operator T

constitute a local operator scheme. Indeed, in order to verify (2) we let

S

i

∈ L(T | E

ⁱ

, F

i

) for i = 1, 2. Given ε > 0, there exist factorizations S

i

= B

i,ε

T A

i,ε

such that kA

i,ε

: E

i

→ Xk ≤ 1 + ε and kB

i,ε

: Y → F

i

k ≤ 1.

Define E to be the direct sum E

₁

⊕ E

2

equipped with the l

₁

-norm, while F is the direct sum F

1

⊕ F

²

under the l

∞

-norm. Put

A

_ε

:= A

_1,ε

Q

^E₁

+ A

_2,ε

Q

^E₂

, B

_ε

:= J

₁^F

B

_1,ε

+ J

₂^F

B

_2,ε

and S

_ε

:= B

_ε

T A

_ε

, where J

_i^E

and Q

^E_i

, J

_i^F

and Q

^F_i

are the canonical injections and surjections, respectively. Then kA

ε

: E → Xk ≤ 1 + ε and kB

ε

: Y → F k ≤ 1. In addition,

S

i

= B

i,ε

T A

i,ε

= Q

^F_i

B

ε

T A

ε

J

_i^E

= Q

^F_i

S

ε

J

_i^E

.

By compactness, the operators S

_ε

∈ (1 + ε)L(T | E, F ) have a cluster point S ∈ L(T | E, F ) as ε → 0. Since S

ⁱ

= Q

^F_i

S

ε

J

_i^E

passes into S

i

= Q

^F_i

SJ

_i^E

, we finally obtain S

i

≤ S for i = 1, 2. Conditions (1) and (3) are obviously fulfilled.

7.4. From the philosophical point of view, the following observation is the most important result of this paper. Operators at hand can be recon- structed from their germs, and finite-dimensional pieces can be glued together to form new operators. Our considerations are based on an isometric con- cept of local equivalence: T

₀ ^loc

∼

1

T

₁

if and only if L(T

0

| E, F ) = L(T

1

| E, F ) for all finite-dimensional test spaces E and F .

Theorem. There is a one-to-one correspondence between

^loc

∼

1

-equiva- lence classes of operators and local operator schemes.

P r o o f. We only need to show that every local operator scheme can be obtained from an operator T , which will be produced by ultraproduct techniques. The underlying index set I consists of all triples i = (S : E → F ) with S ∈ G(E, F ). Condition (2) implies that I is upwards directed. So we can find an ultrafilter U that contains all sections U(i

0

) := {i ∈ I : i ≥ i

0

} with i

0

∈ I. Construct the ultraproducts X := (X

ⁱ

)

^U

, Y := (Y

i

)

^U

and T := (T

i

)

^U

, where X

i

:= E, Y

i

:= F and T

i

:= S.

Fix i

0

= (S

0

: E

0

→ F

0

) ∈ I. If i ∈ U(i

0

), then we find operators A

i

and B

i

such that S

0

= B

i

T

i

A

i

, kA

ⁱ

: E

0

→ X

ⁱ

k ≤ 1 and kB

ⁱ

: Y

i

→ F

⁰

k ≤ 1.

Put A

i

:= O and B

i

:= O whenever i 6∈ U(i

0

). Define

A : e 7→ (A

ⁱ

e)

^U

and B : (y

i

)

^U

7→ U-lim

_i

B

i

y

i

. Then kA : E

0

→ Xk ≤ 1 and kB : Y → F

0

k ≤ 1. Finally,

BT Ae = U-lim

_i

B

i

T

i

A

i

e = S

0

e for all e ∈ E

⁰

.

This means that G(E

0

, F

0

) ⊆ L(T | E

0

, F

0

).

In order to prove the reverse inclusion, let kA : E

⁰

→ Xk ≤ 1 and kB : Y → F

0

k ≤ 1. Write

A =

m

X

h=1

e

^∗_h

⊗ x

h

and B =

n

X

k=1

y

_k^∗

⊗ f

k

,

where x

1

, . . . , x

m

∈ X and y

^∗1

, . . . , y

^∗_n

∈ Y

^∗

are linearly independent. By 5.11, we find an isomorphism I

ε

from F := span {y

1^∗

, . . . , y

^∗_n

} onto some subspace F

ε

of (Y

_i^∗

)

^U

such that kI

^ε

k ≤ 1 + ε and

hT x

h

, y

^∗_k

i = hT x

h

, I

ε

y

^∗_k

i for h = 1, . . . , m and k = 1, . . . , n.

Fix representations x

h

= (x

hi

)

^U

and I

ε

y

_k^∗

= (y

^∗_ki

)

^U

with x

hi

∈ X

i

and y

_ki^∗

∈ Y

i^∗

. In view of Lemma 5.12, there exists U ∈ U such that

m

X

h=1

ξ

_h

x

_hi

≤ (1 + ε)

m

X

h=1

ξ

_h

x

_h

and

n

X

k=1

η

_k

y

_ki^∗

≤ (1 + ε)

n

X

k=1

η

_k

y

^∗_k

for ξ

1

, . . . , ξ

m

∈ K, η

¹

, . . . , η

n

∈ K and i ∈ U. Letting

A

i

:=

m

X

h=1

e

^∗_h

⊗ x

hi

and B

i

:=

n

X

k=1

y

_ki^∗

⊗ f

k

yields operators with kA

i

: E

₀

→ X

i

k ≤ 1 + ε and kB

i

: Y

_i

→ F

0

k ≤ 1 + ε.

Moreover, in view of hT x

h

, y

_k^∗

i = hT x

h

, I

ε

y

^∗_k

i = U-lim

ⁱ

hT

i

x

_hi

, y

^∗_ki

i, it may be achieved that

kBT A − B

ⁱ

T

i

A

i

k =

m

X

h=1 n

X

k=1

( hT x

h

, y

_k^∗

i − hT

i

x

_hi

, y

_ki^∗

i)e

^∗h

⊗ f

k

becomes as small as we please. By (1), we have B

i

T

i

A

i

∈ (1 + ε)

²

G(E

0

, F

0

).

So BT A ∈ G(E

0

, F

0

) as ε → 0, which gives L(T | E

0

, F

0

) ⊆ G(E

0

, F

0

).

7.5. Assigning to (E, F ) the closed unit ball of L(E, F ), we get a local operator scheme. Any generating operator T

0

is maximal with respect to the preordering

^loc

≺; that is, T

^loc

≺ T

0

for all operators T . There is a re- flexive and separable Banach space whose identity map is universal in this sense. Indeed, choosing a dense sequence E

m1

, E

m2

, . . . in the Minkowski compactum of all m-dimensional Banach spaces with m = 1, 2, . . . , we may take the l

2

-sum of the double sequences (E

mn

) so obtained.

8. Injective-local representability

8.1. Let c ≥ 0. We say that an operator T

0

∈ L(X

0

, Y

0

) is injective-

locally c-representable in an operator T

1

∈ L(X

¹

, Y

1

) if for ε > 0, for any

choice of a finite-dimensional space E and N = 1, 2, . . . , for A

0

∈ L(E, X

0

)

and B

0

∈ L(Y

⁰

, l

∞^N

) there exist A

1

∈ L(E, X

¹

) and B

1

∈ L(Y

¹

, l

^N∞

) such that

B

1

T

1

A

1

= B

0

T

0

A

0

and kB

¹

k · kA

¹

k ≤ (c + ε)kB

⁰

k · kA

⁰

k.

In other words, we assume that

L(T

0

| E, l

^N∞

) ⊆ (c + ε)L(T

1

| E, l

^N∞

) whenever ε > 0 or, by 5.10, that

L(T

0

| E, l

∞^N

) ⊆ cL(T

1

| E, l

∞^N

).

This means that, compared with Definition 6.1, we only use a special type of test spaces F . The main point is that l

^N∞

has the metric extension property.

8.2. An operator T

₀

∈ L(X

0

, Y

₀

) is said to be injective-locally repre- sentable in an operator T

1

∈ L(X

¹

, Y

1

) if the above conditions hold for some c ≥ 0. In this case, we write T

^{0 injloc}

≺ T

¹

. Of course,

^injloc

≺ is a preordering on L . Note that T

₀ ^loc

≺ T

1

implies T

₀ ^injloc

≺ T

1

.

8.3. In the following, we need a folklore result.

Lemma. Let ε > 0, A ∈ L(E, X) and B ∈ L(X, l

^∞

(I)). Then there exist B

0

∈ L(X, l

∞^N

) and V ∈ L(l

∞^N

, l

∞

(I)) such that kB

0

k ≤ kBk, kV k ≤ 1 + ε and

E X l

∞

(I)

l

^N_∞

A

//

^B

//

B0



^V

==

In addition, we may arrange that N ≤ (3+2/ε)

ⁿ

, with n := dim(E) replaced by 2n in the complex case.

P r o o f. Choose functionals x

^∗₁

, . . . , x

^∗_N

∈ X

^∗

of norm 1 whose restric- tions to M := A(E) constitute a minimal finite δ-net in the unit sphere of M

^∗

, where δ := ε/(1 + ε). Since (1 − δ)kxk ≤ max

i

|hx, x

^∗i

i| for all x ∈ M, the map B

0

: x 7→ kBk(hx, x

^∗i

i) is invertible on B

⁰

(M ). Finally, V can be obtained as a norm-preserving extension of BB

₀⁻¹

. The estimate of N follows from entropy theory.

8.4. The symbol

^injloc

≺ is justified by the next criterion.

Proposition. Let c ≥ 0. Then the following are equivalent:

(1) T

0

is injective-locally c-representable in T

1

.

(2) T

₀^inj

is locally c-representable in T

₁^inj

.

P r o o f. (1) ⇒(2). Look at the commutative diagram X

0

Y

0

Y

₀^inj

E l

∞^N

G

X

1

Y

1

Y

₁^inj

T0

//

B0



J^inj

Y0

//

C0



A0

OO

A1



V0

==

T1

//

J^inj

Y1

//

B1

OO

V1

``

C1

OO

which is obtained as follows: Let A

0

and C

0

be given. Applying Lemma 8.3 to T

0

A

0

and J

_Y^inj₀

, we find B

0

and V

0

such that kB

0

k ≤ 1 and kV

0

k ≤ 1 + ε.

Choose A

₁

and B

₁

with kB

1

k · kA

1

k ≤ (c + ε)kB

0

k · kA

0

k. Finally, let V

1

be any norm-preserving extension of B

1

and put C

1

:= C

0

V

0

V

1

. Then

kC

1

k · kA

1

k ≤ kC

0

k · kV

0

k · kV

1

k · kA

1

k ≤ (1 + ε)kC

0

k · kB

1

k · kA

1

k

≤ (1 + ε)(c + ε)kC

⁰

k · kA

⁰

k.

(2) ⇒(1). Now we use the commutative diagram X

0

Y

0

Y

₀^inj

E l

_∞^N

X

1

Y

1

Y

₁^inj

T0

//

B0



J^inj

Y0

//

C0

}}

A0

OO

A1



T1

//

J^inj

Y1

//

B1

OO

C1

``

Let A

0

and B

0

be given. Fix any norm-preserving extension C

0

of B

0

. Choose A

1

and C

1

. Finally, let B

1

:= C

1

J

_Y^inj₁

.

8.5. Next, we state a counterpart of 6.8.

Theorem. A property is injective-local if and only if it is injective , ultrapower-stable and global.

8.6. In the case of injective-local representability the list of locally equiv- alent operators, given in 6.3, can be extended by T

^inj

.

Theorem. The operators T , T

^reg

, T

^inj

, T

^∗∗

and T

^U

are injective-locally equivalent.

P r o o f. Note that L(T

^inj

| E, l

^N∞

) = L(T | E, l

∞^N

).

8.7. We now establish an analogue of 6.5.

Proposition. Let c ≥ 0. Then the following are equivalent:

(1) T

0

is injective-locally c-representable in T

1

.

(2) For ε > 0 and for x

01

, . . . , x

0m

∈ X

0

we can find x

11

, . . . , x

1m

∈ X

1

such that

m

X

h=1

ξ

h

x

1h

≤

m

X

h=1

ξ

h

x

0h

and

m

X

h=1

ξ

h

T

0

x

0h

≤ (c + ε)

m

X

h=1

ξ

h

T

1

x

1h

whenever ξ

₁

, . . . , ξ

_m

∈ K.

(3) For ε > 0, for any finite-dimensional space E, and for A

0

∈ L(E, X

0

) there exists A

₁

∈ L(E, X

1

) such that

kA

¹

e k ≤ kA

⁰

e k and kT

⁰

A

0

e k ≤ (c + ε)kT

¹

A

1

e k whenever e ∈ E.

P r o o f. (1) ⇒(2). Let A

⁰

:= J

_M^X0

with M := span {x

⁰¹

, . . . , x

0m

}. Ap- plying 8.3 to T

0

A

0

and J

_Y^inj₀

, we find B

0

and V with J

_Y^inj₀

T

0

A

0

= V B

0

T

0

A

0

, kB

⁰

k ≤ 1 and kV k ≤ 1 + ε. Choose A

¹

∈ L(M, X

¹

) and B

1

∈ L(Y

¹

, l

^N∞

) such that

B

1

T

1

A

1

= B

0

T

0

A

0

, kA

1

k ≤ kA

0

k = 1 and kB

1

k ≤ (c + ε)kB

0

k ≤ c + ε.

Putting x

1h

:= A

1

x

0h

, we finally arrive at

m

X

h=1

ξ

_h

x

_1h

≤

m

X

h=1

ξ

_h

x

_0h

and

m

X

h=1

ξ

h

T

0

x

0h

=

m

X

h=1

ξ

h

J

_Y^inj₀

T

0

A

0

x

0h

≤ kV k ·

m

X

h=1

ξ

h

B

0

T

0

A

0

x

0h

≤ (1 + ε)

m

X

h=1

ξ

_h

B

₁

T

₁

A

₁

x

_0h

≤ (1 + ε)(c + ε)

m

X

h=1

ξ

_h

T

₁

x

_1h

.

(2) ⇒(3). Write A

0

∈ L(E, X

0

) in the form A

0

= P

m

h=1

e

^∗_h

⊗ x

0h

, and let A

1

:= P

m

h=1

e

^∗_h

⊗ x

1h

.

(3) ⇒(1). Given A

0

∈ L(E, X

0

) and B

₀

∈ L(Y

0

, l

^N_∞

), we choose an operator A

1

∈ L(E, X

¹

) such that kA

¹

e k ≤ kA

⁰

e k and kT

⁰

A

0

e k ≤ (c + ε) kT

1

A

1

e k whenever e ∈ E. Of course, kA

1

k ≤ kA

0

k. It follows from

kB

0

T

₀

A

₀

e k ≤ kB

0

k · kT

0

A

₀

e k ≤ (c + ε)kB

0

k · kT

1

A

₁

e k

that T

1

A

1

e 7→ B

⁰

T

0

A

0

e yields a well-defined operator from T

1

A

1

(E) into

l

^N∞

. If B

1

is any norm-preserving extension, then kB

1

k ≤ (c + ε)kB

0

k.

Remark. In the case when T

0

and T

1

are identity maps, the inequalities from (2) pass into

m

X

h=1

ξ

h

x

1h

≤

m

X

h=1

ξ

h

x

0h

≤ (c + ε)

m

X

h=1

ξ

h

x

1h

.

This yields an upper estimate of the Banach–Mazur distance between span {x

01

, . . . , x

0m

} and span{x

11

, . . . , x

1m

}. More precisely, it turns out that the relation

^injloc

≺ extends the concept of crudely finite representability to the setting of operators.

8.8. The next criterion is an analogue of 6.6, from which it could be derived via 8.4. We prefer, however, to give a direct proof.

Proposition. Let c ≥ 0. Then the following are equivalent:

(1) T

0

is injective-locally c-representable in T

1

.

(2) There exist operators A ∈ L(X

⁰

, X

₁^U

) and B ∈ L(Y

1^U

, Y

₀^inj

), where U is an ultrafilter on a suitable index set I, such that

T

₀^inj

= BT

₁^U

A and kBk · kAk ≤ c.

P r o o f. If T

₀^inj

= BT

₁^U

A, then

L(T

0

| E, l

^N∞

) = L(T

0^inj

| E, l

^N∞

) ⊆ kBk · kAkL(T

1^U

| E, l

^N∞

)

⊆ (1 + ε)cL(T

1

| E, l

^N∞

).

This proves that (2) ⇒(1).

We consider the index set I formed by all pairs i = (M, ε). Here M is any finite-dimensional subspace of X

0

and ε > 0. For i = (M, ε) and i

0

= (M

0

, ε

0

), we write i ≥ i

0

if M ⊇ M

0

and 0 < ε ≤ ε

0

. Furthermore, fix some ultrafilter U on I that contains all sections {i ∈ I : i ≥ i

0

} with i

0

∈ I.

Choose A

i

∈ L(M, X

¹

) such that

kA

i

x k ≤ kxk and kT

0

x k ≤ (c + ε)kT

1

A

i

x k for x ∈ M.

Let

x

i

:=  A

i

x if x ∈ M,

o if x 6∈ M, and y

i

:=  T

1

A

i

x if x ∈ M, o if x 6∈ M.

Clearly, A : x 7→ (x

ⁱ

)

^U

defines an operator A ∈ L(X

⁰

, X

₁^U

) with kAk ≤ 1.

Next, it follows from

kT

0^inj

x k = kT

0

x k ≤ U-lim

_i

(c + ε) kT

1

A

_i

x k = ck(y

i

)

^U

k

that B

0

: (y

i

)

^U

7→ T

0^inj

x yields a map from the range of T

₁^U

A into Y

₀^inj

.

Choosing any norm-preserving extension, we obtain the required operator

B ∈ L(Y

1^U

, Y

₀^inj

) with kBk ≤ c. Hence (1)⇒(2).

9. Local and injective-local distances. In the definition of local representability the dimensions of the test spaces E and F are arbitrary.

Now we introduce a gradation.

9.1. For n = 1, 2, . . . , the nth local distance between T

0

and T

1

is defined by

l

_n

(T

₀

, T

₁

) := inf (

c ≥ 0 : L(T

0

| E, F ) ⊆ cL(T

1

| E, F ), dim(E) ≤ n dim(F ) ≤ n

) . If such a constant c ≥ 0 does not exist, we let l

n

(T

0

, T

1

) := ∞. This happens if and only if rank(T

₀

) ≥ n > rank(T

1

). In view of 5.10, we have

l

_n

(T

0

, T

1

) := min (

c ≥ 0 : L(T

0

| E, F ) ⊆ cL(T

1

| E, F ), dim(E) ≤ n dim(F ) ≤ n

) . 9.2. Note that

kT

⁰

k · kT

¹

k

⁻¹

= l

1

(T

0

, T

1

) ≤ l

²

(T

0

, T

1

) ≤ . . . ≤ l

ⁿ

(T

0

, T

1

) ≤ . . . The growth of this sequence measures the deviation of T

1

from T

0

.

For operators T

0

, T

1

and T

2

between arbitrary couples of Banach spaces, we have a multiplicative triangle inequality:

l

n

(T

0

, T

2

) ≤ l

ⁿ

(T

0

, T

1

)l

n

(T

1

, T

2

).

Moreover, 5.6 yields

l

n

(T

₀^∗

, T

₁^∗

) = l

n

(T

0

, T

1

).

9.3. For n = 1, 2, . . . , the nth injective-local distance between T

0

and T

1

is defined by i

_n

(T

0

, T

1

) := inf

(

c ≥ 0 : L(T

0

| E, l

^N∞

) ⊆ cL(T

1

| E, l

^N∞

), dim(E) ≤ n N = 1, 2, . . .

) . If such a constant c ≥ 0 does not exist, we let i

ⁿ

(T

0

, T

1

) := ∞. This happens if and only if rank(T

0

) ≥ n > rank(T

1

). In view of 5.10, we have

i

n

(T

0

, T

1

) := min (

c ≥ 0 : L(T

⁰

| E, l

^∞N

) ⊆ cL(T

¹

| E, l

^∞N

), dim(E) ≤ n N = 1, 2, . . .

) . 9.4. In analogy with 9.2,

kT

0

k · kT

1

k

⁻¹

= i

₁

(T

₀

, T

₁

) ≤ i

2

(T

₀

, T

₁

) ≤ . . . ≤ i

n

(T

₀

, T

₁

) ≤ . . . and

i

_n

(T

₀

, T

₂

) ≤ i

n

(T

₀

, T

₁

)i

_n

(T

₁

, T

₂

).

However, i

n

(T

₀^∗

, T

₁^∗

) and i

n

(T

0

, T

1

) may behave quite differently.

9.5. We now compare the local distances with the injective-local dis- tances.

Proposition. For every ε > 0 there exists a natural number a > 1 such that

i

n

(T

0

, T

1

) ≤ (1 + ε)l

^an

(T

0

, T

1

) for n = 1, 2, . . .

P r o o f. Put a := 4 + [2/ε] (real case) and a := (4 + [2/ε])

²

(complex case). Note that

c

n

:= inf (

c ≥ 0 : L(T

0

| E, l

^N∞⁰

) ⊆ cL(T

1

| E, l

^N∞⁰

), dim(E) ≤ n N

0

≤ a

ⁿ

)

≤ l

aⁿ

(T

0

, T

1

).

We now construct a commutative diagram:

X

₀

Y

₀

E l

∞^N

l

∞^N0

X

1

Y

1

T0

//

B0



C0

A0

OO

A1



oo

V

T1

//

B1

OO

C1

>>

Let δ > 0, A

₀

and B

₀

be given. Applying Lemma 8.3 to T

₀

A

₀

and B

₀

, we find C

0

and V such that kC

⁰

k ≤ kB

⁰

k and kV k ≤ 1 + ε. Choose A

¹

and C

1

with kC

1

k · kA

1

k ≤ (c

n

+ δ) kC

0

k · kA

0

k. Finally, putting B

1

:= V C

1

, we arrive at kB

1

k · kA

1

k ≤ (1 + ε)(c

n

+ δ) kB

0

k · kA

0

k, which proves that

i

n

(T

0

, T

1

) ≤ (1 + ε)(c

n

+ δ) ≤ (1 + ε)(l

aⁿ

(T

0

, T

1

) + δ).

Remark. An inequality of the form i

n

(T

0

, T

1

) ≤ cl

ϕ(n)

(T

0

, T

1

) can only hold if ϕ : N → N grows exponentially: ϕ(n)  a

ⁿ

with a > 1; see 11.6.

9.6. Here is another consequence of Theorem 5.14.

Theorem. The l

_n

-distances between T , T

^reg

, T

^∗∗

and T

^U

equal 1. In the case of i

n

-distances the same is true for T , T

^reg

, T

^inj

, T

^∗∗

and T

^U

.

9.7. Distances between spaces X

0

and X

1

are obtained as the distances between the associated identity maps I

X0

and I

X1

. In the infinite-dimensio- nal case, we have l

n

(X

0

, X

1

) ≤ n and i

ⁿ

(X

0

, X

1

) ≤ √

n. It turns out that i

n

(X

0

, X

1

) is the infimum of all constants c ≥ 1 with the following property:

For every n-dimensional subspace M

0

of X

0

there exists an n-dimensional

subspace M

1

of X

1

such that the Banach–Mazur distance d(M

0

, M

1

) is less

than or equal to c.