Symmetric and Rearrangement- Invariant Sequence Spaces: a Unified Framework

• 1.    Introduction

  In his 1964 work “Embedding Theorems for Banach Spaces of Measurable Functions” [1], E. M. Semenov introduced the concept of a symmetric space: a Banach ideal space E of measurable functions on a ст-finite measure space, such that whenever y G E and | x | is equimeasurable with | y | , one has x G E and ||x|e = ||у||е . When the norm satisfies an additional continuity condition, these spaces, later known as rearrangement-invariant, were previously studied by G. G. Lorentz [2].

• ^# Y. Nessipbayev was partially supported by the grant no. AP26102625 of the Science Committee of the Ministry of Science and Higher Education of the Republic of Kazakhstan. F. Sukochev was partially supported by the Australian Research Council (DP230100434).

• (c) 2026 Zanin, D. V., Nessipbayev, Y. Kh. and Sukochev, F. A.

The notions of symmetric (in the sense of Krein–Petunin–Semenov [8]) and rearrangementinvariant (r. i.) sequence spaces are fundamental concepts in the modern theory of Banach function and sequence spaces (see, e. g., the classical monographs of Bennett–Sharpley [4] and Lindenstrauss–Tzafriri [5] for systematic treatments in the Banach space setting). In the setting of function spaces, these notions form the backbone of studies in interpolation theory and the geometry of Banach lattices.

Traditionally, there are two distinct approaches to the definitions of symmetric sequence spaces. The first approach is based on the usage of the notion of decreasing rearrangement ^ ( x ) of a sequence x, whereas the second approach defines r.i. spaces axiomatically by two natural requirements: invariance under coordinate permutations, and stability with respect to the lattice order. In the setting of Banach function (in particular, sequence) spaces, these two approaches are widely regarded as equivalent and are routinely used interchangeably in the literature. However, even for the purely discrete case of normed sequence spaces, a complete and elementary proof of this equivalence does not appear to have been recorded in the literature. It is the purpose of this note to fill in this gap for a more general case of p-normed sequence spaces.

To motivate the discussion, recall that the l _p -(quasi-)norm when 0 < p <  от (with the usual modification for p = от )

• n       1 /p

( ^ “к)

k =1

of a nonnegative vector ( a i ,..., a n ) depends only on the values { a i ,..., a n } , not on the order of the entries. In this sense, ℓ _p measures the “size” of a vector independently of how its components are arranged. This property—that the (quasi-)norm depends only on the distribution of the sequence values, not their order — is the key idea behind permutationinvariant sequence spaces, which generalize ℓ _p . Rearrangement-invariant function spaces are continuous analogues of these permutation-invariant sequence spaces: in that setting, the (quasi-)norm of a function depends only on the distribution of its values (through the decreasing rearrangement) and not on where those values occur.

Definition 1.1. Let 0 < p C 1 . A p-normed (sequence) space is a linear subspace E C C N (or R N ) equipped with a functional || • ||e : E ^ [0 , от ) satisfying:

• 1.    | x | e = 0 if and only if x = 0 ;

• 2.    ||ax|| E = | a | | x | e for all scalars a and x G E;

• 3. ^{|| x} + y ^ E < llxll E + hhE ^{for al}l x,y ^{G E}-

• If E is complete with respect to the metric d(x,y) = |x — y^E, it is called a p-Banach (sequence) space [6, Chapter 1.3].

Our main result, Theorem 3.9, establishes that in the case of p-normed sequence spaces these two approaches coincide. Specifically, we prove the following theorem

Theorem 1.1. Let ( E, || • | e ) C c o be a p -normed space. Then the following are equivalent:

• (i)    E is rearrangement-invariant ( in the sense of Definition 2 . 1) ;

• (ii)    E is symmetric ( in the sense of Definition 2 . 2) .

• 2.    Preliminaries

  • 2.1.    Notation. We write Z ₊ = { 0 , 1 , 2 ,... } and denote by l ^ the space of bounded real-valued sequences over Z ₊ . For k G Z ₊ , we let e k denote the canonical unit vector.

• 2.2.    Rearrangement-invariant sequence spaces.

Although the equivalence may appear almost self-evident, we emphasize that it has not been explicitly addressed even in standard references. Making this connection precise allows for a unified and flexible language when treating symmetric and rearrangement-invariant sequence spaces. We believe this observation reflects the spirit of the work of E. M. Semenov, who contributed decisively to the structural theory of symmetric and rearrangement-invariant spaces and their role in interpolation and operator theory (see, e.g., the classical book [3], and other works of E. M. Semenov and his collaborators [1; 7–14].

The present note is written in connection with the 85th anniversary of E. M. Semenov. It aims to highlight one of the foundational aspects of symmetric sequence spaces, and to complement the body of results inspired by his deep and substantial contributions to the geometry and interpolation theory of Banach spaces. In the last section we briefly discuss connection of our main result with the definition and properties of symmetrically normed ideals of compact operators.

For x G l _M , we denote by ^ ( x ) its decreasing rearrangement, i. e. the nonincreasing sequence equimeasurable with | x | . To be precise, if x = ( x ( k )) k =o is a bounded sequence of real numbers, then, in what follows, ^ ( x ) = ( ^ ( k,x )) ^ g denotes the “decreasing rearrangement” of the sequence | x | = ( | x ( k ) | ) k =o . That is, it is defined by the formula

^ ( k,x ) =    inf sup | x ( i ) | , k G Z + .

A⊂Z+ i∈Ac card(A)=k

Note that, in general, it is not quite the decreasing permutation of the sequence | x | . For example, let x = (1 — y +y ) k =o , then ^ ( x ) = (1 , 1 ,..., 1 ,... ) . However, in cases where the sequence | x | can be arranged in a non-increasing fashion (e. g., when it has only finitely many values or decreases monotonically), ^ ( x ) coincides with the usual decreasing permutation, as the name “decreasing rearrangement” suggests.

Definition 2.1. A p-normed space ( E, || • ||e ) with E C l _№ is called rearrangementinvariant ( r.i. ), if

• (i)    for every x G E and every permutation (bijection) n : Z ₊ ^ Z ₊ we have x о n G E and | x о п | е = | x | e (permutation/rearrangement-invariance);

• (ii)    if x G E and y G l _№ with | y | C | x | , then y G E and | у | е C | x | e (lattice stability).

• 2.3.    Symmetric sequence spaces.

Definition 2.2. A p-normed space ( E, || • ||e ) with E C l _№ is called symmetric, if

• (i)    for every x G E and y G l _№ with ^ ( x ) = ^ ( y ) we have y G E and |x|e = ||у||е ;

• (ii)    for every x G E and y G l _№ with | y | C | x | we have y G E and ||y ||e C | x | e .

• 3.    Proof of Theorem 1.1

Conditions (i) and (ii) in the latter definition are equivalent to the following single condition: if x G E and ^ ( y ) C ^ ( x ) , then y G E and | у | е C | x | e [3, Definition 1, II.4].

These two definitions form a foundation for the so-called Calkin correspondence , which provides a link between symmetric sequence spaces and two-sided ideals of compact operators. This correspondence has been developed and extensively studied in many works, including the monographs [15, 16], the comprehensive treatment by Krein, Petunin, and Semenov [3], and more recently in the monograph by Dodds, de Pagter, and Sukochev [17].

This section is devoted to proving Theorem 1.1. Specifically, the forthcoming lemmas establish the implication (i) ^ (ii).

Lemma 3.1. Let ( E, | • | e ) C c o be an r.i. p -normed space. If there exists 0 = x G E, then e k G E for every k ^ 0 and | e ^ | e = | e o | E for every k ^ 0 .

The proof is not difficult, so we omit it. In what follows, we assume without loss of generality that ||eo|| E = 1 .

It should be noted that ^ ( x ) need not be a permutation of x, even if x G c o . Nonetheless, the situation in c o is more controlled. That is, one can always relate ^ ( x ) to a rearrangement of x via a suitable permutation, as made precise in the next lemma.

Lemma 3.2. If 0 C x G c o , then there exists a permutation n : Z ₊ ^ Z ₊ , such that exactly one of the following ( mutually exclusive ) options holds.

• ( i) ^ ( x ) = x о n ;

• ( ii) x ( n (2 k )) = ^ ( k,x) >  0 and x ( n (2 k + 1)) = 0 for every k ^ 0;

• ( iii) x ( n ( k + n )) = ^ ( k,x) >  0 for every k ^ 0 and x ( n ( k )) = 0 for 0 C k < n for some

n G N .

• < 1 Set A := { k G Z . : x ( k ) >  0 } and B := { k G Z . :   x ( k ) = 0 } . If B = 0, then

^(x) = x о n for some permutation n and so case (i) holds while cases (ii) and (iii) are not possible. The same is true for every finite set A. Hence, we may assume that A is infinite and B is non-empty (either finite or infinite).

Suppose that B is infinite. Cases (i) and (iii) are not possible. Fix e >  0 . Choose a permutation n, such that n : 2 Z ₊ ^ A and n : Z ₊ \ 2 Z ₊ ^ B. Replacing x with x о n if needed, we may assume that A = 2 Z ₊ and B = Z ₊ \ 2 Z ₊ . Choose a permutation n, such that n = id on Z ₊ \ 2 Z ₊ and such that ^ ( k, x ) = x ( n (2 k )) for every k ^ 0 . Replacing x with x о n if needed, we may assume without loss of generality that x (2 k ) = ^ ( k, x ) >  0 and x (2 k +1) = 0 for every k ^ 0 , i. e. case (ii) holds.

Finally, suppose B is a finite set. Cases (i) and (ii) are not possible. Let |B| = n for some n G N. Choose a permutation n such that n : {0, • • • ,n — 1} ^ B. Replacing x with x о n if needed, we may assume without loss of generality that B = {0, • • • ,n — 1}. Choose a permutation n, such that n = id on {0, • • • ,n — 1} and such that ^(k,x) = x(n(k + n)) for every k ^ 0. Replacing x with x о n if needed, we conclude that x(k + n) = ^(k, x) > 0 for every k ^ 0 and x(k) = 0 for every 0 C k < n, that is, case (iii) holds. >

We now show that every element of an r. i. sequence space has its decreasing rearrangement in the space, with (quasi-)norm no greater than the original. This is a fundamental property of r. i. spaces and will play a key role in proving symmetry.

Lemma 3.3. Let ( E, | • | e ) C c o be an r. i. p -normed space. If x G E, then ^ ( x ) G E and h^{( x ) |} E ^C llxll E .

• <    Without loss of generality, assume x ^ 0 . Let n be a permutation as in Lemma 3.2. By assumption, we have x о n G E and | x о п | е = | x | e . Therefore, replacing x with x о n if needed, we may assume that n = id . Thus, as in Lemma 3.2 we consider the following three cases.

Case (i): We have ^ ( x ) = x. In this case the claim is trivial.

Case (ii): We have x (2 k ) = ^ ( k,x ) >  0 and x (2 k + 1) = 0 for every k ^ 0 .

Define y G c o by y (2 k + 1) = ^ ( k, x ) and y (2 k ) = 0 for every k ^ 0 . Let n be a permutation such that n : 2 k О 2 k + 1 for every k ^ 0 so that y = x о n. Since E is r. i., it follows that y G E and 1 у 1 е = | x | e .

Let e >  0 . Set z = x + ey. Since x,y G E, it follows from the p-sub-additivity of | • ||e that z G E and | z | E C (1 + e^p ) | x | E . Therefore,

He c (1+ ^ZhIe .

Note z(k) > 0 for every k, therefore, ^(z) = z о n‘ for some permutation n‘. Hence, ^(z) G E and |^(z)|e = |z|e C (1 + ep)1/p||x||e. Since ^(x) C ^(z), it follows from lattice stability that ^(x) G E and ||д(х)||e < ||Xz)||e C (1 + Ep)i/pHx\e. Since e > 0 is arbitrary, the claim follows.

Case (iii): We have x ( k + n ) = ^ ( k, x ) >  0 for every k ^ 0 and x ( k ) = 0 for 0 C k < n for some fixed n G N .

Let e > 0. Set z = x + EX[o,n). Since ek G E by Lemma 3.1 and \ek\e = 1 for every k ^ 0, it follows from the p-sub-additivity that X[0,n) G E and that ||X[0,n) He C ni/p. Since x,X[o,n) G E, it follows from the p-sub-additivity that z G E and ||z||E C HxHE + Epn. Hence, \\z\\e c (HxHE + Epn)1/p.

Since z ( k) >  0 for every k, it follows that ^ ( z ) = z о n ‘ for some permutation n ‘ . Hence, ^ ( z ) G E and H ^ ( z ) H e = \ z \ e C ( H x H E + E^pn)^1/p. Clearly, ^ ( x ) C ^ ( z ) . Thus, ^ ( x ) G E and |G( x )| e C ( H x H E + E^pn)^1/p. Since n is fixed and e >  0 is arbitrary, we have that |G( x )| e C \ x \ e . >

To establish the converse statement (see Lemma below) we need several auxiliary lemmas.

Lemma 3.4. Let ( E, 11 •H e ) C c o be an r. i. p -normed space. If x G c o is such that ^ ( x ) G E, then x G E. Moreover, for every A C Z . with | A | = | A^c | = от we have

H x H E c H m ( x ) xa H E + H ^ ( x ) x a c H E .

• < 1 Without loss of generality, assume x ^ 0 . Let A C Z . be such that | A | = | A^c | = от and let n be a permutation as in Lemma 3.2. By assumption, we have x о n G E and H x о п | е = | x | e • Therefore, replacing x with x ◦ n if needed, we may assume that n = id . As in Lemma 3.2 we consider the following three cases.

Case (i): Assume ^ ( x ) = x. In this case the claim is obvious.

Case (ii): Assume x (2 k ) = ^ ( k,x ) >  0 and x (2 k + 1) = 0 for every k ^ 0 .

By lattice stability, we have ^(x)xa G E and ^(x)xac G E. Choose permutations ni and П2, such that ni : 2k ^ k for every k G A and such that П2 : 2k ^ k for every k G Ac. We have xX2A = (Mx)XA) о П1, xX2^c = (Xx)XAc) о П2, where xx2A is a componentwise product of two sequences x and X2A (with 1’s at position k for every k G 2A and 0’s otherwise). Indeed, ni(2A) = A and so if n G 2A i. e. n = 2a for some a G A, then (xx2A)(n) = x(n) = x(2a) = ^(a,x) = (Xx)XA)(a) = (Mx)XA)(ni(2a)) = (Mx)XA) оni)(n). If n G 2A, then ni(n) G A, so both sides are zeros. Similarly, for the second equality.

Since ( E, H • || e ) is r.i., it follows that xx ₂ A ,xx₂- A c G E with | xx ₂ a | _E = IIMx)xa H e and H xx 2 A c H e = HX x ) X A c H e . Since x is supported on even indices (i. e. x = xx ₂ A + xx ₂ A ), it follows that x G E and from p -triangle inequality we have

||x \ E ^{C H} xX 2 A ^H E + ^{H x} X 2 A \ E = IM^{x )} X A ^H E + IH^{x )} X A c H E .

Case (iii): Assume there n G N such that x ( k + n ) = ^ ( k, x ) for every k ^ 0 and x ( k ) = 0 for 0 C k < n.

Since ^(x) G E by lattice stability we have ^(x)xa G E and ^(x)xac G E. Choose permutations ni and П2, such that ni : k + n ^ k for every k G A and such that П2 : k + n ^ k for every k G Ac . We have xXn+A = (Mx)XA) о ni, xXn+Ac = (^(x)xAc) о П2.

Since (E, || • ||e) is r.i., we have xXn+A,xXn+Ac € E with ^xXn+AHe = ^^(x)XAHe and ^xXn+Ac He = b(x)XAc He• By p-triangle inequality, we have x € E and llxllE C ||xXn+A WE + WxXn+Ac WE C IHx)XA|lE + IMx)XAc HE, which completes the proof. О

We state the following immediate corollary of Lemma 3.4 for future reference.

Corollary 3.1. Let ( E, || • | e ) C c o be an r.i. p -normed space. If x € c o is such that ^ ( x ) € E, then x € E and H x H e C 2 ^{1 /p} • H ^ ( x ) H e •

The next lemma shows that for a decreasing sequence in an r.i. space, one can isolate an infinite subset of coordinates on which the (quasi-)norm becomes arbitrarily small.

Lemma 3.5. Let ( E, || • | e ) C c o be an r.i. p -normed space. If x = ^ ( x ) € E, then for every e >  0 there exists A C Z + , such that | A | = | A^c | = от and H xx a H e C e H x H e •

• < 1 The statement is clear if x has a finite support. Assume x = ^ ( x ) € E has infinite support and let e >  0 .

Choose a strictly increasing sequence { k ( m ) } m ^ o of even numbers, such that x ( k ( m)) C ^ ly p x ( m ) for all m ^ 0 . Set A = { k ( m ) : m ^ 0 } . We have ^ ( xx a ) C ^ r /p ^ ( x ) . Applying Corollary 3.5 to xχ A , we obtain

HxxaHpe c 2|Mxxa)IIpe c 2 • EpHm(x)WE = ep||x|IE, hence, |xxa|e C e|x|e. О

Now we are ready to obtain the converse result to Lemma 3.3 and strengthen the estimate from Corollary 3.5.

Lemma 3.6. Let ( E, W • H e ) C c o be an r.i. p -normed space. If x € c o and if ^ ( x) € E, then x € E and H x H e ^C IH^x)IIe .

• <    Let x € c o be such that ^ ( x ) € E. By Lemma 3.4, x € E. Let e >  0 and A C Z . be as in Lemma 3.5 (applied to ^ ( x ) ). By Lemma 3.4 and Lemma 3.5, we have

W x W E C IHx ) x aII ^pe + IMx ) xa c H ^pe C (1 + e^p ) H m ( x ) H E .

Since e >  0 is arbitrary, the proof is complete. О

Proposition 3.1. Every r. i. p -normed space ( E, 11 • H e ) C c o is symmetric. Namely, let x be in an r.i. p -normed space E and y be in l ^ such that ^ ( y ) = ^ ( x ) . Then y € E and ^Н У ^Н Е = ||^xW E .

• <    It follows from Lemma 3.3 that ^ ( x ) € E and ||^ ( x ) W E C H x H e - It follows from Lemma 3.6 that H x H e ^C IM^x)IIe . That is, ^ ( x ) € E and ||^(x)He = H x H e ■ Therefore, ^ ( y ) € E (since ^ ( x ) = ^ ( y ) ) and ||^(у)||е = H x H e ■ Note y € c o since x € c o and ^ ( x ) = ^ ( y ) . It follows from Lemma 3.6 that y € E and Н у Н е C Н м ( у ) H e = H x H e ■ It follows from Lemma 3.3 that Н у Н е >  1Ы у ) Н е = H x H e . Thus, y € E and Н у Н е = H x H e . О

Now we are ready to prove the main result of this paper.

Theorem 3.1. Let ( E, | • H e ) C c o be a p -normed space. Then the following are equivalent:

• (i)    E is rearrangement-invariant ( in the sense of Definition 2 . 1);

• (ii)    E is symmetric ( in the sense of Definition 2 . 2) .

• <    The implication (i) = ^ (ii) follows from Proposition 3.1. The implication (ii) = ^ (i) follows from Definitions 2.1 and 2.2. Indeed, let x € E and n be a permutation of Z ₊ . It is clear that ^ ( x ) = ^ ( x о n ) . By definition of a symmetric space we have x о n € E and II^xIIe = IM^x)IIe = IHx ◦ ^п)|е = ^{| x} ◦ ^п|е . ^o

• 3.1.    Space of bounded sequences ℓ _∞ . So far, we have established the equivalence of the definitions for r. i. and symmetric p-normed sequence spaces contained in c 0 . The following lemma shows that this assumption is not overly restrictive: in fact, the only r. i. (or symmetric) sequence space not contained in c 0 is ℓ _∞ itself.

Proposition 3.2. Let ( E, \\ • 11 e ) C l ^ be an r. i. p -normed space. If E C c o , then E = l ^ ( as sets' ) and \| • \ e is equivalent to \| • ||_ю .

• <1 Let z G E \ c o . Without loss of generality, z ^ 0 . Choose e >  0 , such that the set A := { n G Z + : z ( n ) ^ e } is infinite. Since z ^ ex A , it follows that X A G E. Choose an infinite set B C A, such that B ^c is also infinite. By lattice stability we have x b G E. Choose a permutation n such that x b c = X B ◦ n. Since E is r. i., it follows that x b c G E and the constant sequence 1 = x b + X B c G E. By linearity it follows that l ^ C E and, therefore, E = l ^ .

• 3.2.    Ideals of compact operators. We conclude with a short discussion of the notions of r. i. and symmetric spaces in the non-commutative setting of ideals of compact operators.

For every x G l ^ , we have 0 C | x ( n ) | С |Ик for all n G Z + . Thus, ||x||e C \ x U\¹\e . On the other hand, we have | x | ^ | x ( k ) | e k for all k G Z + . Thus, ||x||e ^ | x ( k ) || e k |e = | x ( k ) l\ e o \ e . Taking the supremum over k ^ 0 , we obtain ||x||e ^ ||x | _M \ e o \ E • Thus, || • \ e is equivalent to | • | ^ . >

We could not prove or disprove the equivalence of these two definitions in the case E = l ^ , as well as for general quasi-normed sequence spaces (although every quasi-norm is equivalent to some p -norm by the Aoki–Rolewicz theorem [6, Theorem 1.3]). Hence, we finish this section with the following question.

Question. Does the equivalence between (i) and (ii) from Theorem 3.1 hold for all quasinormed sequence spaces (not necessarily p -normed), and in particular, is every permutationinvariant norm on ℓ _∞ (respecting the lattice stability) necessarily symmetric?

Let E C c o be a symmetric (equivalently, r. i.) Banach sequence space equipped with the norm || • | e and H be a separable Hilbert space. Define the associated (to E ) space C e of compact operators as follows

Ce := {A G K(H) : ^(A) G E}, where ^(A) = (^(n, A))n^o are the singular values of A, that is, the eigenvalues of the positive operator |A| = (A*A)1/2 arranged in non-increasing order and counted with multiplicities. Note that in this case µ is also called a generalized singular value sequence and is given by [20, p. 89]:

^ ( k, A ) = inf { | A (1 — p ) | ^ : p G B ( H ) , rank( p ) C k } , k ^ 0 .

Also, C e is a two-sided ideal in B ( H ) [16, Corollary 2.3.17]. It is shown in [20] that when equipped with the norm

|A\ce := HA)||e, the space CE becomes a Banach space. Moreover, by [16, Theorem 3.1.1(c)], it follows that Ce is symmetric (or symmetric operator space) in the sense that if A G Ce and B G B(H) with ^(B) C ^(A) then B G Ce and \B\ce C \A\ce [16, Definition 2.5.1], compare with Definition 2.2. By [17, Proposition 4.4.3], it also has the following property (which is also sometimes called ‘symmetric’ [16, Definition 1.2.12])

\\ XAY ||ce C | X IUIAIIce | Y U A G C e , X,Y G B ( H ) .

In this setting, the role of permutations is replaced by conjugation by unitaries. Indeed, let H = I 2 , with the canonical orthonormal basis ( e _n ) _n ^ i . Given a sequence x = ( x n ) G 1 ^ , consider the diagonal operator

D x := diag( x i ,x 2 ,... ) G B ( I 2 )

Then for any permutation n of N , define the unitary operator U _n in B ( l 2 ) by U _n e n := e _n ( n ) , n G N . Straightforward computation shows that U _n D x U n = D x o_n . Thus, permuting coordinates of a sequence by π corresponds exactly to unitary conjugation by unitaries U _π . Hence, it is natural to call C e rearrangement-invariant (r. i.) (or rather the norm || • ||c_e is r. i.), if

| UAV ||c_e = ||A||c_e , A G C E , U,V unitaries in B ( H ) .

A classical example is the Schatten p-class: for E = I p , C e = S _p with norm (for more general examples see [20])

/ ^          \ i/p lAk =

E Kn,A^p) , n =0

1 <  p <  x .

Theorem 3.11. Let ( E, || • |e ) be a rearrangement-invariant space. If E C c o , then C e is a normed ideal in B ( H ) with unitarily invariant norm

I UA I c e = l AU I c e = I|A||ce , A G C e , U unitary in B ( H ) .

• <1 By Theorem 1.2, ( E, || • | e ) is not only rearrangement-invariant, but also symmetric. The assertion follows now from [20]. In particular, C e is both r. i. and symmetric. >

Moreover, if E is Banach, then C E admits an equivalent norm that is monotone with respect to the Weyl (logarithmic) submajorization.

Definition 3.1 (Weyl or logarithmic submajorization). Let x,y G 1 ^ (or B ( H ) ). If for all n ^ 0 , nn

П ^ ^{( k,}y ) <  ПЕ^к,х) , k =0           k =0

then y is said to be logarithmically submajorized by x, written y ^^ log x.

It follows from [19, Theorem 7] that every Banach ideal in B ( H ) (in particular, C e ) and every symmetric Banach sequence space admits an equivalent norm monotone with respect to ^^ log . We record this as an explicit, straightforward consequence.

Proposition 3.3. Let E C c o be a symmetric Banach sequence space, and let || • | E be an equivalent norm on E monotone under logarithmic submajorization:

y ^log x ^ IlyHE < llxllE , ^Х,У ^G E.

Define for A G C e ,

I A I c e := I ^ ( A ) I ' e .

Then || • | C is an equivalent norm on C e , monotone with respect to ^^ log : if ^ ( B) ^^ log ^ ( A ) , then I B I C e <  I A I C e . Hence every symmetric Banach sequence space admitting a ^^ log - monotone norm induces such a norm on C E .

• <1 Since || • Ц Е and || • Ц е are equivalent, there exist c,C> 0 such that с Ц х Ц е C ||x H E ^ C Ц х Ц е for all x € E. Thus, for any A € C e ,

сЦАЦсе = c|HA)IIe C IHA)HE = Wce c CЦ^(А)Це = сЦАЦсе, so Ц • Ice is equivalent to Ц • Ice•

As ^ ( B ) ^^k> _g ^ ( A ) , we have

ЦвICe = Ыв)Ц‘Е c Ц^(А)ЦЕ = ЦАЦСе, as required. >