## Bounded Operators and Isomorphisms of Cartesian Products of Fréchet Spaces

### P. D ja k ov, T. T e r z i o ˘g lu, M . Yu r da k u l ,

### & V. Z a har i u ta

**Introduction**

### In [25; 26] it was discovered that there exist pairs of wide classes of Köthe spaces *(X , Y ) such that*

*L(X, Y ) = LB(X, Y ) if X ∈ X , Y ∈ Y,* (1) where *LB(X, Y ) is the subspace of all bounded operators from X to Y. If either* any *X ∈ X is Schwartzian or any Y ∈ Y is Montel, then this relation coincides* with

*L(X, Y ) = L* _{c} *(X, Y ) if X ∈ X , Y ∈ Y,* (2) where *L* _{c} *(X, Y ) denotes the subspace of all compact operators.*

_{c}

_{c}

### This phenomenon was studied later by many authors (see e.g. [1; 5; 11; 12; 13;

### 14; 15; 20; 21]); of prime importance are Vogt’s results [24] giving a generally complete description of the relations (1) for the general case of Fréchet spaces (for further generalizations see also [3; 4]).

### Originally, the main goal in [25; 26] was the isomorphism of Cartesian prod- ucts (and, consequently, the quasi-equivalence property for those spaces). The pa- pers made use of the fact that, due to Fredholm operators theory, an isomorphism of spaces *X × Y ' X* 1 *× Y* 1 *(X, X* 1 *∈ X , Y, Y* 1 *∈ Y ) that satisfies (2) also implies* an isomorphism of Cartesian factors “up to some finite-dimensional subspace”.

### In the present paper we generalize this approach onto classes *X × Y of prod-* ucts that satisfy (1) instead of (2). Although Fredholm operators theory fails, we have established that—in the case of Köthe spaces—the stability of an automor- phism under a bounded perturbation still takes place, but in a weakened form: “up to some basic Banach space”. In particular, we get a positive answer to Question 2 in [7]: Is it possible to modify somehow the method developed in [25; 26] in or- der to obtain isomorphic classification of the spaces *E* 0 *(a) × E*

_{∞}

*(b) in terms of* sequences *a, b if a* _{i} *6→ ∞ and b* _{i} 6→ ∞?

_{i}

_{i}

### Some of our results are announced without proofs in [9].

### Received October 16, 1997. Revision received May 28, 1998.

### Research of the first author was supported by the TÜB˙ITAK–NATO Fellowship Program and partially by the NRF of Bulgaria, grant no. MM-808/98.

### Michigan Math. J. 45 (1998).

### 599

**1. Preliminaries**

### Let *(a* *ik* *)* *i,k∈N* be a matrix of real numbers such that 0 *≤ a* *ik* *≤ a* *i,k+1* for all *i, k.*

### We denote by *K* ^{p} *(a* _{ik} *), 1 ≤ p < ∞, the l* ^{p} -Köthe space defined by the matrix *(a* _{ik} *)—that is, the space of all sequences of scalars x = (x* _{i} *) such that*

^{p}

_{ik}

^{p}

_{ik}

_{i}

*|x|* _{k} : = X

_{k}

*i*

*(|x* _{i} *|a* _{ik} *)* ^{p}

_{i}

_{ik}

^{p}

### 1 */p*

*< ∞ ∀k ∈ N;*

### with the topology generated by the system of seminorms {| · | *k* *, k ∈ N}, it is a* Fréchet space. If *a = (a* *i* *) is a sequence of positive numbers, then the Köthe* spaces

*E* _{0} ^{p} *(a) = K* ^{p}

^{p}

^{p}

### exp

### − 1 *k* *a* _{i}

_{i}

*,* *E* ^{p}

^{p}

_{∞}

*(a) = K* ^{p} *(exp{ka* _{i} *})*

^{p}

_{i}

### are called, respectively, *l* ^{p} -finite and *l* ^{p} -infinite type power series spaces. These spaces are Schwartz if and only if *a* *i* *→ ∞. If (a* *i* *) and (b* *i* *) are sequences of* positive numbers such that *a* *i* *→ ∞ and b* *i* *→ ∞, then E* ^{p} 0 *(a) ' E* ^{p} 0 *(b) (resp.*

^{p}

^{p}

^{p}

^{p}

*E*

∞^{p} *(a) ' E*

∞^{p}

^{p} *(b)) if and only if a* *i* * b* *i* *, that is,*

^{p}

*∃c > 0 : a* _{i} */c ≤ b* _{i} *≤ c ∀i* (see [18]).

_{i}

_{i}

### Each Köthe space has a natural basis *(e* _{j} *), where e* _{j} *= (δ* _{ji} *). A subspace gen-* *erated by the subsequence of the natural basis is called basic subspace. It is well* known that *K* ^{p} *(a* *ik* *) is not a Montel space if and only if there exist k* 0 and a sub- sequence of indices *(i* *n* *) such that*

_{j}

_{j}

_{ji}

^{p}

*∀k ∃C* _{k} : *a* _{i}

_{k}

_{i}

_{n}_{k} *≤ C* _{k} *a* _{i}

_{k}

_{k}

_{i}

_{n}_{k}

0 _{k}

*∀n.*

### Therefore we have the following proposition.

### Proposition 1. *An* *l* ^{p} *-Köthe space is non-Montel if and only if it contains a basic* *subspace isomorphic to* *l* ^{p} *.*

^{p}

^{p}

### If *X and Y are topological vector spaces then a linear operator T : X → Y is* *bounded (resp. compact) if there exists a neighborhood of zero* *U in X such that* *T (U) is a bounded (precompact) set in Y. We write (X, Y ) ∈ B (resp. (X, Y ) ∈ K)* if each continuous linear operator from *X into Y is bounded (resp. compact).*

### We say that a pair *(X, Y ) has the bounded (resp. compact) factorization prop-* erty and write *(X, Y ) ∈ BF (resp. (X, Y ) ∈ KF ) if each linear continuous oper-* ator *T : X → X that factors through Y (i.e. T = S* 1 *S* 2 *, where S* 2 : *X → Y and* *S* 1 : *Y → X are linear continuous operators) is bounded (resp. compact).*

### After Dragilev [10] and Bessaga [2], a Köthe matrix *(a* *ik* *) is said to be of type* *(d* 1 *) or (d* 2 *) if, respectively, the following condition holds:*

*(d* 1 *) ∃k* 0 *∀k ∃(m, C) : a* _{ik} ^{2} *≤ Ca* _{ik}

0_{ik}

_{ik}

*a* _{im} *,*

_{im}

*(d* 2 *) ∀k ∃m ∀l ∃C : a* *ik* *a* *il* *≤ Ca* ^{2} _{im} *.*

_{im}

### Let *E be a Fréchet space with basis (e* *i* *) and fundamental system of seminorms* *(| · |* *k* *). If for some p ∈ [1, ∞) we have*

*x =* X

*i*

*x* *i* *e* *i* ⇒ X

*(|x* *i* *||e* *i* | *k* *)* ^{p} *< ∞ ∀k,*

^{p}

### then the basis *(e* _{i} *) is called l* ^{p} *-absolute and* *E is isomorphic to the l* ^{p} -Köthe space *K* ^{p} *(|e* _{i} | _{k} *). If the corresponding Köthe matrix is of type (d* 1 *) or (d* 2 *) then we say* that *E is a (d* 1 *)- or (d* 2 *)-space and write E ∈ (d* 1 *) or E ∈ (d* 2 *). Recall that finite* (resp. infinite) type power series spaces are *(d* 2 *) (resp. (d* 1 *)) spaces.*

_{i}

^{p}

^{p}

^{p}

_{i}

_{k}

### Zahariuta [26] showed that *(X, Y ) ∈ B if X and Y are locally convex spaces,* with *l* ^{1} -absolute bases, satisfying respectively the conditions *(d* 2 *) and (d* 1 *). By the* results of Vogt [24] (see Satz 6.2 and Prop. 5.3) it follows that the same is true for spaces with *l* ^{p} -absolute basis, so the following proposition holds.

^{p}

### Proposition 2. *If* *X is (d* 2 *)-Köthe space and Y is (d* 1 *)-Köthe space then (X, Y ) ∈* *B. In particular, for any p, q ∈ [1, ∞) we have (E* ^{p} _{0} *(a), E*

^{p}

_{∞}

^{q} *(b)) ∈ B.*

^{q}

**2. Bounded Operators in Köthe Spaces**

### The following statement is crucial for our approach.

### Lemma 1. *If* *X = K(a* *ik* *) is a Köthe space and A ⊂ X is a bounded set, then* *for any* *k* 0 *and any* *ε > 0 there exists a Banach basic subspace B such that A ⊂* *B + εU* *k*

0*, where U* *k*

0*= { x ∈ X : |x|* *k*

0 *≤ 1}.*

*Proof. We give the proof for* *l* ^{1} -Köthe spaces; the case *p > 1 can be treated* similarly. Since the set *A is bounded we may assume without loss of generality* that

*A =*

*x ∈ X : |x|* _{k} = X

_{k}

*i*

*a* _{ik} *|x* _{i} *| ≤ C* _{k} *∀k*

_{ik}

_{i}

_{k}

### *.*

### Choose *C* *k* *% ∞ so that a* *ik* */C* *k* *→ 0 for all i. Set γ* *i* = P

*k* *(a* *ik* */2* ^{k} *C* *k* *); then* X

^{k}

*i*

*γ* *i* *|x* *i* | = X

*i*

### X

*k*

*a* _{ik} 2 ^{k} *C* _{k}

_{ik}

^{k}

_{k}

*|x* *i* | = X

*k*

### 1 2 ^{k}

^{k}

### X

*i*

*a* _{ik} *C* _{k} *|x* *i* |

_{ik}

_{k}

### ≤ 1 for any *x ∈ A. Fix any ε > 0 and set*

*B = [e* _{i} : *εγ* _{i} *≤ a* _{ik}

0_{i}

_{i}

_{ik}

### ] *,* *E = [e* _{i} : *εγ* _{i} *> a* _{ik}

0_{i}

_{i}

_{ik}

### ] *,*

### where the square brackets denote the closed linear span of the corresponding vec- tors. Then obviously *B is a Banach space and for x ∈ A ∩ E we have*

*|x|* *k*

0### = X

*i*

*a* *ik*

0*|x* *i* *| < ε* X

*i*

*γ* *i* *|x* *i* *| < ε,*

### which proves the statement.

### Remark 1. It is easy to see that, under the assumptions and notations of the

### lemma, if the set *A is compact then for any k* 0 and for any *ε there exists a fi-*

### nite-dimensional basic subspace *B such that A ⊂ B + εU* *k*

0*.*

### Theorem 1. *If* *X is a Köthe space and T : X → X is a bounded (resp. compact)* *operator, then there exist complementary basic subspaces* *B and E such that:*

### (i) *B is a Banach (resp. finite-dimensional) space; and*

*(ii) if* *π* _{E} *and* *i* _{E} *are the canonical projection onto* *E and embedding into X,* *respectively, then the operator 1* _{E} *− π* *E* *Ti* *E* *is an automorphism on* *E.*

_{E}

_{E}

_{E}

*Proof. Suppose* | · | _{p} *, p ∈ N, is a fundamental system of norms in X. Since T is a* bounded operator there exists a *k* 0 such that *T (U* _{k}

0_{p}

_{k}

*) is a bounded set in X. Hence*

*∀k ∃C* _{k} : *|Tx|* _{k} *≤ C* _{k} *|x|* _{k}

0_{k}

_{k}

_{k}

_{k}

*.*

### By Lemma 1 (resp. Remark 1) there exists a Banach (resp. finite-dimensional) basic subspace *B such that T (U* _{k}

0_{k}

*) ⊂ B +* ^{1} _{2} *U* _{k}

0_{k}

*. Let E be the basic subspace that* is complementary to *B. Then, setting T* 1 *= π* _{E} *Ti* _{E} : *E → E, we obtain that*

_{E}

_{E}

*|T* 1 *x|* *k*

0 ### ≤ ^{1} _{2} *|x|* *k*

0 *∀x ∈ E.*

### Now it is easy to see that the operator 1 _{E} *− T* 1 is an automorphism. Indeed, for any *x ∈ E consider the series*

_{E}

*Sx = x + T* 1 *x + T* 1 ^{2} *x + · · · + T* 1 ^{m} *x + · · · .* (3) This series is convergent in *E because, for any k, we have*

^{m}

*|T* 1 ^{m} *x|* *k* *≤ C* *k* *|T* 1 ^{m−1} *x|* *k*

0 ^{m}

^{m−1}

*≤ C* *k* ¡ _{1}

### 2

### _{m−1}

_{m−1}

*|x|* *k*

0*, m = 1, 2, . . .,*

### and so, by the Banach–Steinhaus theorem, (3) defines a linear continuous opera- tor *S : E → E. Since (1* _{E} *− T* 1 *)Sx = S(1* _{E} *− T* 1 *)x = x, the operator S is inverse* to the operator 1 _{E} *− T* 1 *.*

_{E}

_{E}

_{E}

**3. Isomorphisms of Cartesian Products**

### As usual, we identify an operator *T : E* 1 *× E* 2 *→ F* 1 *× F* 2 with the corresponding 2 *× 2 matrix (T* *ij* *), whose entries are operators acting between the factors of the* Cartesian products.

### Lemma 2. *Let* *E* 1 *, E* 2 *, F* 1 *, F* 2 *be topological vector spaces. If* *T = (T* *ij* *): E* 1 × *E* 2 *→ F* 1 *×F* 2 *is an isomorphism such that* *T* 11 : *E* 1 *→ F* 1 *is also an isomorphism,* *then* *E* 2 *' F* 2 *.*

*Proof. Let* *T*

^{−1}

*= (S* _{ij} *). Consider the operators*

_{ij}

*S* 22 : *F* 2 *→ E* 2 *,* *H : E* 2 *→ F* 2 *,*

### where *H = T* 22 *− T* 21 *T* 11

^{−1}

*T* 12 *. Taking into account that T* 11 *S* 12 *+ T* 12 *S* 22 *= 0, we* obtain

*HS* 22 *= T* 22 *S* 22 *− T* 21 *T* 11

^{−1}

*T* 12 *S* 22 *= T* 22 *S* 22 *+ T* 21 *S* 12 = 1 *F*

2*.*

### In an analogous way, from *S* 21 *T* 11 *+ S* 22 *T* 21 = 0 it follows that

*S* 22 *H = S* 22 *T* 22 *− S* 22 *T* 21 *T* 11

^{−1}

*T* 12 *= S* 22 *T* 22 *+ S* 21 *T* 12 = 1 *E*

2*.* Hence the spaces *E* 2 and *F* 2 are isomorphic.

### The next theorem is a modification of the generalized Douady lemma in [26, Sec.

### 6]. In [8] an analogous modification is obtained by considering Riesz type opera- tors instead of bounded operators.

### Theorem 2. *Suppose* *X* 1 *is a Köthe space and* *X* 2 *, Y* 1 *, Y* 2 *are topological vector* *spaces. If* *X* 1 *× X* 2 *' Y* 1 *× Y* 2 *and* *(X* 1 *, Y* 2 *) ∈ BF, then there exist complemen-* *tary basic subspaces* *E and B in X* 1 *and complementary subspaces* *F and G in Y* 1

*such that* *B is a Banach space and*

*F ' E,* *B × X* 2 *' G × Y* 2 *.* *If, in addition,* *(Y* 1 *, X* 2 *) ∈ BF, then G is a Banach space.*

*Proof. Let* *T = (T* *ij* *): X* 1 *× X* 2 *→ Y* 1 *× Y* 2 be an isomorphism, and let *T*

^{−1}

### = *(S* *ij* *). Then we have S* 11 *T* 11 *+ S* 12 *T* 21 = 1 *X*

1*. Since the operator S* 12 *T* 21 is bounded, by Theorem 1 there exist complementary basic subspaces *E and B of X* 1 such that *B is a Banach space and the operator A = π* *E* *S* 11 *T* 11 *i* *E* is an automorphism of *E.*

### It is easy to see that the operator *P = T* 11 *A*

^{−1}

*π* *E* *S* 11 is a projection on *Y* 1 *. We set* *F = P (Y* 1 *),* *G = P*

^{−1}

*(0).*

### Obviously we have *F = T* 11 *(E) and, moreover, the restriction of T* 11 on *E is an* isomorphism between *E and F. From Lemma 2 it now follows that B × X* 2 ' *G × Y* 2 *.*

### If, in addition, each operator acting in *Y* 1 that factors through *X* 2 is bounded, then the same is true for each operator acting in *G that factors through X* 2 *. Suppose* *H : G×Y* 2 *→ B ×X* 2 is an isomorphism and let *(H* *ij* *) and (R* *ij* *) be operator 2×2* matrices corresponding to *H and H*

^{−1}

*. Then we have 1* _{G} *= R* 11 *H* 11 *+ R* 12 *H* 21 *.* Here the operator *R* 12 *H* 21 is bounded because it factors through *X* 2 and the oper- ator *R* 11 *H* 11 is bounded because it factors through the Banach space *B. Hence the* operator 1 _{G} is bounded; that is, *G is a Banach space.*

_{G}

_{G}

### Remark 2. One can easily see by the proof and by Theorem 1 that: (a) if *(X* 1 *, Y* 2 *) ∈ KF then the space B may be chosen to be finite-dimensional; and* (b) if, in addition, *(Y* 1 *, X* 2 *) ∈ KF then the space G also will be finite-dimensional.*

### So, in this case we obtain a statement that is known (see [8; 26]).

**4. Applications**

### We begin with an observation showing that an infinite-dimensional complemented Banach subspace in an *l* ^{p} -Köthe space is isomorphic to *l* ^{p} *.*

^{p}

^{p}

### Proposition 3. *Let* *X be an l* ^{p} *-Köthe space, and let* *F and G be complementary*

^{p}

*subspaces in* *X (i.e., X = F ⊕ G). If G is an infinite-dimensional Banach space*

*then* *G ' l* ^{p} *and, moreover,* *F and G are isomorphic to some basic subspaces of X.*

^{p}

*Proof. We have* *X × {0} ' F × G. By Theorem 2 there exist complementary ba-* sic subspaces *E and B in X and complementary subspaces F* 1 and *G* 1 in *F such* that *B is a Banach space and*

*F* 1 *' E,* *B ' G* 1 *× G.*

### Since every infinite-dimensional basic Banach subspace of an *l* ^{p} -Köthe space is isomorphic to *l* ^{p} *, we obtain that B ' l* ^{p} *. On the other hand, each infinite-* dimensional complemented subspace of *l* ^{p} *, 1 ≤ p < ∞, is isomorphic to l* ^{p} (see [22] or [16]), so *G is isomorphic to l* ^{p} *. Finally, since B ' l* ^{p} *, its complemented* subspace *G* 1 is isomorphic to some basic subspace of *B and F ' E ⊕ G* 1 is iso- morphic to some basic subspace of *X.*

^{p}

^{p}

^{p}

^{p}

^{p}

^{p}

^{p}

### This result may be considered as a partial answer to the well-known Pelczyn- ski problem: Does a complemented subspace of a space with basis have a ba- sis? Moreover, in this case we confirm the conjecture of Bessaga [2] that each complemented subspace of a Köthe space is isomorphic to a basic subspace.

### The following theorem answers Question 2 in [7]. In fact, we consider a more general situation.

### Theorem 3. *Suppose* *X* 1 *, X* 2 *, Y* 1 *, Y* 2 *are non-Montel* *l* ^{p} *-Köthe spaces such that* *X* 1 *× X* 2 *' Y* 1 *× Y* 2 *. If X* 1 *, Y* 1 *∈ (d* 2 *) and X* 2 *, Y* 2 *∈ (d* 1 *) then X* 1 *' Y* 1 *and* *X* 2 *' Y* 2 *.* *Proof. By Proposition 2, each operator acting in* *X* 1 (resp. *Y* 1 *) that factors through* *Y* 2 (resp. *X* 2 *) is bounded. Thus, by Theorem 2 there exist complementary basic* subspaces *E and B in X* 1 and complementary subspaces *F and G in Y* 1 such that

^{p}

*F ' E,* *B × X* 2 *' G × Y* 2 *,*

### and *B and G are Banach spaces. Then B (resp. G) is either a finite-dimensional* space or (by Proposition 3) isomorphic to *l* ^{p} *.*

^{p}

### Obviously, since *l* ^{p} *× l* ^{p} *' l* ^{p} *, we have B × l* ^{p} *' l* ^{p} and *G × l* ^{p} *' l* ^{p} *. From* here and Proposition 1 it follows immediately that

^{p}

^{p}

^{p}

^{p}

^{p}

^{p}

^{p}

*X* 1 *' X* 1 *× l* ^{p} *' E × B × l* ^{p} *' F × G × l* ^{p} *' Y* 1 *× l* ^{p} *' Y* 1

^{p}

^{p}

^{p}

^{p}

### and

*X* 2 *' X* 2 *× l* ^{p} *' X* 2 *× B × l* ^{p} *' Y* 2 *× G × l* ^{p} *' Y* 2 *× l* ^{p} *' Y* 2 *.*

^{p}

^{p}

^{p}

^{p}

### In [8], the isomorphic classification of Cartesian products *E* ^{p} _{0} *(a) × E*

^{p}

_{∞}

^{q} *(b) was* studied by using strictly singular operators. Necessary and sufficient conditions were obtained for the isomorphism

^{q}

*E* ^{p} 0 *(a) × E*

^{p}

_{∞}

^{q} *(b) ' E* 0

^{q}

^{˜p}*(˜a) × E*

_{∞}

^{˜q}*( ˜b)*

### in the case where *p 6= ˜q or q 6= ˜p. However the approach used in [8] does not*

### work in the case where *p = ˜q and q = ˜p. The previous theorem covers the case*

*p = q = ˜p = ˜q; the case where p 6= q, ˜q = p, and ˜p = q is treated in the next*

### theorem. We consider only the non-Montel case, since if some of the spaces are

### Montel then the result is known by [26].

### Theorem 4. *Suppose* *p 6= q and the spaces E* 0 ^{p} *(a), E*

∞^{p}

^{q} *(b), E* 0 ^{q} *(˜a), E*

∞^{q}

^{q}

^{p} *( ˜b) are* *non-Montel. Then the following conditions are equivalent.*

^{p}

### (i) *E* ^{p} 0 *(a) × E*

∞^{p}

^{q} *(b) ' E* 0 ^{q} *(˜a) × E*

∞^{q}

^{q}

^{p} *( ˜b).*

^{p}

### (ii) *E* ^{p} _{0} *(a) × E*

∞^{p}

^{q} *(b)* ^{qd} *' E* _{0} ^{q} *(˜a) × E*

∞^{q}

^{q}

^{p} *( ˜b) (where “qd” denotes quasi-diagonal).*

^{p}

*(iii) There exist complementary subsequences* *a*

^{0}

*, a*

^{00}

*, b*

^{0}

*, b*

^{00}

*, ˜a*

^{0}

*, ˜a*

^{00}

*, ˜b*

^{0}

*, ˜b*

^{00}

*respec-* *tively of* *a, b, ˜a, ˜b such that a*

^{00}

*, b*

^{00}

*, ˜a*

^{00}

*, ˜b*

^{00}

*are bounded;*

*E* ^{p} 0 *(a*

^{p}

^{00}

*) ' l* ^{p} *, E* 0 ^{q} *(˜a*

^{p}

^{q}

^{00}

*) ' l* ^{q} *, E*

^{q}

_{∞}

^{q} *(b*

^{q}

^{00}

*) ' l* ^{q} *, E*

^{q}

_{∞}

^{p} *( ˜b*

^{p}

^{00}

*) ' l* ^{p} ; *the spaces* *E* ^{p} _{0} *(a*

^{p}

^{p}

^{0}

*), E*

_{∞}

^{q} *(b*

^{q}

^{0}

*), E* ^{q} _{0} *(˜a*

^{q}

^{0}

*), E*

_{∞}

^{p} *( ˜b*

^{p}

^{0}

*) are nuclear; and a*

^{0}

_{i} * ˜a*

_{i}

^{0}

_{i} *and* *b* _{i}

_{i}

_{i}

^{0}

* ˜b* _{i}

_{i}

^{0}

*. That is,*

*E* ^{p} 0 *(a*

^{p}

^{0}

*) ' E* 0 ^{q} *(˜a*

^{q}

^{0}

*),* *E*

_{∞}

^{q} *(b*

^{q}

^{0}

*) ' E*

_{∞}

^{p} *( ˜b*

^{p}

^{0}

*).*

*Proof. Since (iii)* *⇒ (ii) ⇒ (i) we prove only that (i) ⇒ (iii). If (i) holds then,* by Proposition 2 and Theorem 2, there exist complementary subsequences *a*

^{0}

### and *a*

^{00}

### of *a and complementary subspaces F* 1 and *G* 1 of *E* _{0} ^{q} *(˜a) such that E* ^{p} _{0} *(a*

^{q}

^{p}

^{00}

*) and* *G* 1 are Banach spaces and

*E* ^{p} _{0} *(a*

^{p}

^{0}

*) ' F* 1 *,* *E* ^{p} _{0} *(a*

^{p}

^{00}

*) × E*

_{∞}

^{q} *(b) ' G* 1 *× E*

^{q}

_{∞}

^{p} *( ˜b).*

^{p}

### By Proposition 3 there exist complementary subsequences *˜a*

^{0}

### and *˜a*

^{00}

### of *˜a such that* *F* 1 *' E* 0 ^{q} *(˜a*

^{q}

^{0}

*), G* 1 *' E* 0 ^{q} *(˜a*

^{q}

^{00}

*), E* ^{p} 0 *(a*

^{p}

^{00}

*) is either finite-dimensional or isomorphic to* *l* ^{p} *, and E* _{0} ^{q} *(˜a*

^{p}

^{q}

^{00}

*) is either finite-dimensional or isomorphic to l* ^{q} *. Then E* _{0} ^{p} *(a*

^{q}

^{p}

^{0}

*) '* *E* 0 ^{q} *(˜a*

^{q}

^{0}

*), so by [8, Prop. 4] these spaces are nuclear. From here it follows that* the spaces *E* ^{p} 0 *(a*

^{p}

^{00}

*) and E* 0 ^{q} *(˜a*

^{q}

^{00}

*) are infinite-dimensional because otherwise E* ^{p} 0 *(a)* or *E* _{0} ^{q} *(˜a) would be nuclear (hence Montel). By Mityagin’s characterization of* isomorphic power series spaces, we obtain that *a*

^{p}

^{q}

^{0}

_{i} * ˜a*

_{i}

^{0}

_{i} *.*

_{i}

### Thus we have

*E*

_{∞}

^{q} *(b) × l* ^{p} *' E*

^{q}

^{p}

_{∞}

^{p} *( ˜b) × l* ^{q} *.*

^{p}

^{q}

### Now by Theorem 2 there exist complementary subsequences *b*

^{0}

### and *b*

^{00}

### of *b and* complementary subspaces *F* 2 and *G* 2 in *E*

∞^{p} *( ˜b) such that E*

∞^{p}

^{q} *(b*

^{q}

^{00}

*) and G* 2 are Ba- nach spaces and *E*

∞^{q} *(b*

^{q}

^{0}

*) ' F* 2 *. Using Proposition 3, we obtain that there exist* complementary subsequences ˜ *b*

^{0}

### and ˜ *b*

^{00}

### of ˜ *b such that*

*E*

_{∞}

^{p} *( ˜b*

^{p}

^{0}

*) ' F* 2 *,* *E* ^{p}

^{p}

_{∞}

*( ˜b*

^{00}

*) ' G* 2 *.*

### Now from the same argument as before it follows that the isomorphic spaces *E*

_{∞}

^{q} *(b*

^{q}

^{0}

*) and E*

_{∞}

^{p} *( ˜b*

^{p}

^{0}

*) are nuclear, and Mityagin’s characterization of isomorphic* power series spaces shows that *b* _{i}

_{i}

^{0}

* ˜b* _{i}

_{i}

^{0}

*. Finally by Proposition 3 we have*

*E*

_{∞}

^{q} *(b*

^{q}

^{00}

*) ' l* ^{q} and *E*

^{q}

_{∞}

^{p} *( ˜b*

^{p}

^{00}

*) ' l* ^{p} *.*

^{p}

### The methods presented here and in [8] may be used to study the isomorphic classification of the Cartesian products *K* ^{p} *(A) × K* ^{q} *(B), where A is (d* 2 *)-matrix* and *B is (d* 1 *)-matrix. One can easily generalize the results of [8] in order to obtain* characterizations of isomorphisms

^{p}

^{q}

*K* ^{p} *(A) × K* ^{q} *(B) ' K*

^{p}

^{q}

^{˜p}*( ˜* *A) × K*

^{˜q}*( ˜B)*

### in the case where *p 6= ˜q or q 6= ˜p. In fact, our Theorem 3 treats the case p =* *q = ˜p = ˜q (which is impossible to treat with the methods of [8]; see Question 2* in [7]). The next theorem is the corresponding generalization of Theorem 4.

### Theorem 5. *Let* *p 6= q. Suppose that K* ^{p} *(A) and K* ^{q} *( ˜* *A) are non-Montel (d* 2 *)-* *Köthe spaces, and that* *K* ^{q} *(B) and K* ^{p} *( ˜B) are non-Montel (d* 1 *)-Köthe spaces.*

^{p}

^{q}

^{q}

^{p}

*Then the following statements are equivalent.*

### (i) *K* ^{p} *(A) × K* ^{q} *(B) ' K* ^{q} *( ˜* *A) × K* ^{p} *( ˜B).*

^{p}

^{q}

^{q}

^{p}

*(ii) There exist complementary submatrices* *A*

^{0}

*, A*

^{00}

*, B*

^{0}

*, B*

^{00}

*, ˜* *A*

^{0}

*, ˜* *A*

^{00}

*, ˜B*

^{0}

*, ˜B*

^{00}

*re-* *spectively of* *A, B, ˜* *A, ˜B such that*

*K* ^{p} *(A*

^{p}

^{00}

*) ' l* ^{p} *, K* ^{q} *( ˜* *A*

^{p}

^{q}

^{00}

*) ' l* ^{q} *, K* ^{q} *(B*

^{q}

^{q}

^{00}

*) ' l* ^{q} *, K* ^{p} *( ˜B*

^{q}

^{p}

^{00}

*) ' l* ^{p} ; *the spaces* *K* ^{p} *(A*

^{p}

^{p}

^{0}

*), K* ^{q} *(B*

^{q}

^{0}

*), K* ^{q} *( ˜* *A*

^{q}

^{0}

*), K* ^{p} *( ˜B*

^{p}

^{0}

*) are nuclear; and*

*K* ^{p} *(A*

^{p}

^{0}

*) ' K* ^{q} *( ˜* *A*

^{q}

^{0}

*),* *K* ^{q} *(B*

^{q}

^{0}

*) ' K* ^{p} *( ˜B*

^{p}

^{0}

*).*

*Proof. Since obviously (ii)* *⇒ (i), we need only prove that (i) ⇒ (ii). If (i) holds* then, by Proposition 2 and Theorem 2, there exist complementary submatrices *A*

^{0}

### and *A*

^{00}

### of *A and complementary subspaces F* 1 and *G* 1 of *K* ^{q} *( ˜* *A) such that K* ^{p} *(A*

^{q}

^{p}

^{00}

*)* and *G* 1 are Banach spaces and

*K* ^{p} *(A*

^{p}

^{0}

*) ' F* 1 *,* *K* ^{p} *(A*

^{p}

^{00}

*) × K* ^{q} *(B) ' G* 1 *× K* ^{p} *( ˜B).*

^{q}

^{p}

### By Proposition 3 there exist complementary submatrices ˜ *A*

^{0}

### and ˜ *A*

^{00}

### of ˜ *A such that* *F* 1 *' K* ^{q} *( ˜* *A*

^{q}

^{0}

*), G* 1 *' K* ^{q} *( ˜* *A*

^{q}

^{00}

*), K* ^{p} *(A*

^{p}

^{00}

*) is either finite-dimensional or isomorphic* to *l* ^{p} *, and K* ^{q} *( ˜* *A*

^{p}

^{q}

^{00}

*) is either finite-dimensional or isomorphic to l* ^{q} *. Then K* ^{p} *(A*

^{q}

^{p}

^{0}

*) '* *K* ^{q} *( ˜* *A*

^{q}

^{0}

*), so by [8, Prop. 4] these spaces are nuclear. From here it follows that the* spaces *K* ^{p} *(A*

^{p}

^{00}

*) and K* ^{q} *( ˜* *A*

^{q}

^{00}

*) are infinite-dimensional because otherwise K* ^{p} *(A) or* *K* ^{q} *( ˜* *A) would be nuclear (hence Montel).*

^{p}

^{q}

### Now we have

*K* ^{q} *(B) × l* ^{p} *' K* ^{p} *( ˜B) × l* ^{q} *.*

^{q}

^{p}

^{p}

^{q}

### Repeating the same argument as before, we obtain that there exist complementary submatrices *B*

^{0}

*, B*

^{00}

### of *B and ˜B*

^{0}

*, ˜B*

^{00}

### of ˜ *B such that*

*K* ^{q} *(B*

^{q}

^{0}

*) ' K* ^{p} *( ˜B*

^{p}

^{0}

*), K* ^{q} *(B*

^{q}

^{00}

*) ' l* ^{q} *, K* ^{p} *( ˜B*

^{q}

^{p}

^{00}

*) ' l* ^{p} *,* and the spaces *K* ^{q} *(B*

^{p}

^{q}

^{0}

*) and K* ^{p} *( ˜B*

^{p}

^{0}

*) are nuclear.*

### Let us note that, in [8] and in Theorem 4, a stronger result was proved: Carte- sian products of power series spaces may be isomorphic if and only if they are quasi-diagonally isomorphic. The proof was based on Mityagin’s results [18; 19]

### that two power series spaces are isomorphic if and only if they are quasi-diagonally

### isomorphic. In general, it is an open problem whether *(d* 1 *)- and (d* 2 *)-Köthe spaces*

### have this property.

**5. Generalizations and Comments**

### In the previous section we consider applications to the isomorphic classification of Cartesian products of *(d* 1 *)- and (d* 2 *)-spaces. Further applications may be obtained* by using the results of Vogt [24] concerning the relation *(X, Y ) ∈ B for Fréchet* spaces. Namely, Vogt proved that if *X and Y are Fréchet spaces such that X has* the property *(LB*

^{∞}

*) and Y has the property (DN ), then each operator from X to* *Y is bounded. We refer to [24] for definitions of the properties (LB*

^{∞}

*) and (DN )* for Fréchet spaces. Here we note only that for Köthe spaces the property *(DN )* is equivalent to the property *(d* 1 *) and, by [24, Prop. 5.4], it is known that a Köthe* space generated by a matrix *(a* *ik* *) has the property (LB*

^{∞}

*) if and only if*

*∀ρ* _{k} *↑ ∞ ∀p ∃q ∀n* 0 *∃(N* 0 *, C) ∀i ∃k, n* 0 *≤ k ≤ N* 0 : *a* _{ik} *a* _{ip} ^{ρ}

_{k}

_{ik}

_{ip}

^{ρ}

^{k}*≤ Ca* ^{1} _{iq}

_{iq}

^{+ρ}^{k}*.* Obviously, it is possible to generalize Theorem 3 and Theorem 5 by considering Köthe spaces with the property *(LB*

^{∞}

*) instead of (d* 2 *)-Köthe spaces.*

### There are other wide classes of Fréchet spaces for which it is possible to apply the results of Section 3. Recall that a Fréchet space *X is called a quojection if, for* any continuous seminorm *q(·) on X the quotient space X/Ker q is Banach. We* refer to the survey [17] for details concerning quojections.

### From [3, 23] it is known that if *E is a quojection then (E, F ) ∈ B if and only* if *F has a continuous norm. Using this fact, we immediately obtain the following* statement from Theorem 2.

### Theorem 6. *Suppose* *E* 1 *, E* 2 *are quojections and* *F* 1 *, F* 2 *are Köthe spaces ad-* *mitting continuous norms. If* *E* 1 *× F* 1 *' E* 2 *× F* 2 *, then there exist complementary* *basic subspaces* *B* 1 *, H* 1 *in* *F* 1 *and* *B* 2 *, H* 2 *in* *F* 2 *such that* *B* 1 *and* *B* 2 *are Banach* *spaces,* *H* 1 *' H* 2 *, and B* 1 *× E* 1 *' B* 2 *× E* 2 *.*

### We may generalize Theorem 6 by considering prequojections instead of quojec- tions. Recall that a Fréchet space *E is called prequojection if its bidual space E*

^{00}

### is a quojection. Each quojection is a prequojection. It is known (see [17; 23]) that *(E, F ) ∈ B if E is a prequojection and F is a Fréchet space with continuous norm* and the bounded approximation property. Hence, in Theorem 6 we may replace the requirement “ *E* 1 *, E* 2 are quojections” by “ *E* 1 *, E* 2 are prequojections”.

### We suspect that the results of [3; 4] may be used to obtain further generaliza- tions.

### In all applications we consider, in fact we used the relation *B instead of the* weaker relation *BF. It is easy to give an example of a nontrivial pair (E, F ) with* the property *BF.*

### Example. First we note that if *E, F* 1 *, F* 2 are Fréchet spaces such that *(E, F* 1 *) ∈*

*B and (F* 2 *, E) ∈ B, then obviously we have that each operator, acting in E, that*

### factors through *F* 1 *× F* 2 is bounded. We choose *E, F* 1 *, F* 2 to be appropriate Drag-

### ilev *L* *f* -spaces. Recall that if *f is a logarithmically convex function and (a* *i* *) is*

### a sequence of real numbers such that *a* *i* *↑ ∞, then the corresponding Dragilev*

### space of infinite type is defined as

*L* *f* *((a* *i* *), ∞) = K(exp f(ka* *i* *)).*

### Let *f* 1 *, f, f* 2 be chosen in such a way that the functions *f* _{1}

^{−1}

*◦ f and f*

^{−1}

*B f* 2 are rapidly increasing. We put

*E = L* _{f} *((a* _{i} *), ∞),* *F* 1 *= L* _{f}

1_{f}

_{i}

_{f}

*((a* _{i} *), ∞),* *F* 2 *= L* _{f}

2_{i}

_{f}

*((a* _{i} *), ∞).*

_{i}

### Then it is known by [14] that *(F* 1 *, E) ∈ K and (E, F* 2 *) ∈ K but that (E, F* 1 *) /∈ K* and *(F* 2 *, E) /∈ K. Setting F = F* 1 *× F* 2 we obtain the desired example.

### One may therefore expect further applications provided the following problem is solved.

### Problem 1. Characterize pairs of Fréchet spaces *(E, F ) with the property BF.*

### Our crucial argument was the observation, stated in Lemma 1, that each bounded set in a Köthe space is “small up to a complemented Banach subspace”. This ar- gument was used to prove Theorem 1. Let us consider the following generaliza- tion of this property. We say that a Fréchet space *X with a fundamental system of* seminorms *(| · |* _{k} *) has the property (SCBS) if, for any bounded set A ⊂ X and for* any *k* 0 and *ε > 0, there exist complementary subspaces B and E in X such that* *B is a Banach space and*

_{k}

*A ⊂ B + εU* _{k}

0_{k}

*∩ E.* (4)

### It is easy to see that the class of Fréchet spaces with property (SCBS) is larger than the class of Köthe spaces. Recall that if *(a* *ik* *) is a Köthe matrix and E* *i* is a se- quence of Banach spaces then we can consider the corresponding “Banach-valued Köthe space”

*X = { x = (x* *i* *) : x* *i* *∈ E* *i* *, kxk* *k* = P

*i* *a* *ik* *|x* *i* | *i* *< ∞ ∀k },*

### where | · | *i* is the norm in *E* *i* *. Equipped with the system of seminorms (k · k* *k* *), X* is a Fréchet space.

### Proposition 4. *Each Banach-valued Köthe space has the property (SCBS ).*

### The proof is the same as for Lemma 1.

### Now, repeating with slight changes the proof of Theorem 1, we obtain the fol- lowing generalization.

### Theorem 7. *If* *X is a Fréchet space with property (SCBS) and T : X → X is a* *bounded operator, then there exist complementary subspaces* *B and E such that:*

### (i) *B is a Banach space; and*

*(ii) if* *π* _{E} *and* *i* _{E} *are the canonical projection onto* *E and embedding into X,* *respectively, then the operator 1* _{E} *− π* _{E} *Ti* _{E} *is an automorphism on* *E.*

_{E}

_{E}

_{E}

_{E}

_{E}