3. Multi-bounded operators
3.1. Definitions and basic results
indeed,Tpnqis specified by the formula
Tpnq:px
1, . . . , xnq ÞÑ pT x1, . . . , T xnq, En ÑFn. Suppose that pEn,
} ¨ }nq and pFn,} ¨ }nq are two power-normed spaces, and that T PBpE, Fq. It follows from (2.1.1) that the nth amplification of T is bounded as a
linear map frompEn,} ¨ }nq to pFn,} ¨ }nq(with }T} ď››Tpnq ›
›ďn}T}) for eachn PN.
However, in general, the norms ››Tpnq ›
› will not be uniformly bounded in n P N. The
following generalizes definitions given in [20,§6.1.3]. Recall that βpSqis the embedding constant of an operatorS, as on page 12.
Definition 3.1. Let pEn,} ¨ }nq and pFn,} ¨ }nqbe power-normed spaces, and suppose thatT PBpE, Fq. ThenT ismulti-bounded, with norm}T}mb, if
}T}mb:“sup !› › ›T pnq›› ›:nPN ) ă 8.
The mapT is a multi-contraction, respectively, amulti-isometry, if the map Tpnq:pEn,
} ¨ }nq Ñ pFn,} ¨ }nq
is a contraction, respectively, an isometry, for eachnPN. Further,T is amulti-isomor-
phism if it is a bijection and if both T :E ÑF and T´1 :F
ÑE are multi-bounded, andT is amulti-embedding if it is an embedding and if inftβpTpnqq:nP
Nu ą0. The spaces pEn,
} ¨ }nq and pFn,} ¨ }nq are multi-isomorphic, respectively, multi-iso-
metric, if there is a multi-isomorphism, respectively, a bijective multi-isometry fromE ontoF.
The collection of multi-bounded maps fromE toF is denoted by MpE, Fq.
In particular, in the case wherepEn,} ¨ }nqandpFn,} ¨ }nqarep–multi-normed spaces for some p with 1 ď p ď 8, we shall sometimes say that T is p–multi-bounded if it is multi-bounded with respect to the two p–multi-norms, and we shall write MppE, Fq for the collection of p–multi-bounded maps from E to F. In this case, the norm of a p–multi-bounded operatorT PMppE, Fqis sometimes denoted by}T}p´mb.
In the case wherepEn,
} ¨ }nqandpFn,} ¨ }nqare operator sequence spaces, our defin- itions coincide with those of sequentially bounded maps, sequential contractions, and
For a study ofM8pE, FqandM8pEq “M8pE, Eq(in the setting of multi-bounded spaces), see [20, Chapter 6].
LetEandF be Banach spaces such thatpEn,
} ¨ }nqandpFn,
} ¨ }nqare power-normed spaces. ThenpMpE, Fq,} ¨ }mbqis easily seen to be a Banach space;cf.[20, Theorem 6.15].
Example 3.2. Takep, q such that 1ďp, q ď 8, suppose thatpEn,} ¨ }nqis ap–multi- normed space and that pFn,
} ¨ }nq is a q–multi-normed space, and consider the space pMpE, Fq,} ¨ }mbq. We suppose thatE, F ‰ t0u.
First suppose thatpďq. TakeyPF andλPE1, and considerT :“ybλPFpE, Fq. Then, fornPNandx“ px1, . . . , xnq PEn, we have
› › ›T pnqx›› › n
“ }pxx1, λyy, . . . ,xxn, λyyq}n
“ ˜ n ÿ j“1 |xxj, λy|q ¸1{q }y} by equation (2.2.4) ď ˜ n ÿ j“1 |xxj, λy|p ¸1{p }y}
ďµp,npxq }λ} }y} ď }x}n}λ} }y} by Theorem 2.11,
and so T P MpE, Fq with }T}mb “ }λ} }y}. It follows that FpE, Fq Ă MpE, Fq. In particular,MpE, Fq ‰ t0u.
Second suppose thatpąq. Take TPBpE, FqwithT ‰0, and then takexPE with }x} “1 and T x ‰0. For nP N, set x “ px, . . . , xq PEn. By (2.2.5), }x}n “n1{p and
› ›Tpnqx › › n “ }T x}n 1{q, and so › ›Tpnq › › ě }T x}n1{q´1{p Ñ 8 as n Ñ 8. It follows that
T RMpE, Fq, and soMpE, Fq “ t0u.
Example 3.3. Let E and F be Banach spaces, and take p, q with 1 ď p ď q ă 8. Consider the weak p–summing norm pµp,nq based on E, so that pµp,nq is a strong p–multi-norm, and the q–sum power-norm p} ¨ }`q
npFqq based on F, so that p} ¨ }`nqpFqq
is a power-norm that is sometimes a (strong) q–multi-norm. Then the space of multi- bounded operators frompEn, µ
p,nqtopFn, } ¨ }`q
npFqqwith the multi-bounded norm} ¨ }mb
is exactly the spacepΠq,ppE, Fq, πq,pqofpq, pq–summing operators fromE toF, and so }T}mb“πq,ppTq pT PΠq,ppE, Fqq.
Consider the special case whenF “E andq“p; we shall write ΠppEqfor Πp,ppE, Eq, πp forπp,p, andπppEqforπppIEq, as is standard. Thus
› › ›IE :pE n, µp,nq Ñ pEn, } ¨ }`p npEqq › › ›mb“πppEq. (3.1.1)
See [20,§3.4.2], for example, for background onpq, pq–summing operators. LetpEn,
} ¨ }nqbe a power-normed space, and letF be a linear subspace ofE. Then the inclusion JF : F Ñ E is a multi-isometry. Suppose that F is closed in E. As we remarked after equation (1.3.5), for eachnPN, we identify thenth amplification of the quotient mapping QF : E ÑE{F with the quotient mapping of En onto En{Fn, and soQF is a multi-contraction.
LetpEn,
} ¨ }nqand pFn,
} ¨ }nq be two power-normed spaces, and take T P BpE, Fq and n P N. Recall from equation (1.3.20) that we have identified the dual of the nth amplification ofTwith thenthamplification of the dualT1ofT, so thatpTpnqq1“ pT1qpnq. Moreover, we have identified thenth amplification of the canonical embedding ofEinto its bidualE2with the canonical embedding ofEninto its bidualpEn
q2. Since the latter operator is an isometry, we see that the canonical embedding of a power-normed space into its bidual is a multi-isometry.
Proposition 3.4. Let pEn,} ¨ }nq and pFn,} ¨ }nq be power-normed spaces, and take
T PBpE, Fq. Then T is multi-bounded if and only if T1 : F1 Ñ E1 is multi-bounded
with respect to the dual power-norms based onF1 andE1, respectively, and, in this case, }T1}
mb“ }T}mb. In the case wherepEn,} ¨ }nqandpFn,} ¨ }nqarep–multi-normed spaces
for somepwith1ďpď 8andT PMppE, Fq, we have T1 PM
p1pF1, E1q.
Proof. TakenPN. Then››pT1qpnq › ›“ › ›pTpnqq1 › ›“ › ›Tpnq ›
›. ThusT1is multi-bounded if and
only ifT is multi-bounded; in this case,}T1}
mb“ }T}mb.
The following remarks are contained in [20, Chapter 6] in the setting of multi-norms, but they apply in the setting of power-norms and, in particular, forp–multi-norms.
Definition 3.5. LetpEn,} ¨ }nq be a power-normed space, and takepxiq PEN. Then pxiqis amulti-null sequence inE if, for each εą0, there exists n0PNsuch that
sup kPN
}pxn`1, . . . , xn`kq}k ăε pněn0q.
LetpEn,
} ¨ }nqandpFn,
} ¨ }nqbe two power-normed spaces, and takeT PBpE, Fq. Then T ismulti-continuous ifpT xiqis a multi-null sequence inF wheneverpxiqis a multi-null sequence inE.
The following result has the same proof as [20, Theorem 6.14].
Theorem 3.6. Let pEn,} ¨ }nq and pFn,} ¨ }nq be two power-normed spaces, and take
T PBpE, Fq. ThenT is multi-continuous if and only if T is multi-bounded.
We shall next prove the power-normed analogue of the theorem on quotient operators stated as Proposition 1.4(i). This result will be used later.
Proposition 3.7. Let pEn,} ¨ }nq andpFn,} ¨ }nqbe power-normed spaces, and suppose thatT PBpE, Fq. Then the operatorT:E{kerT ÑF induced byT is a multi-isometry if and only ifTpnq is a quotient operator for eachnP
N.
Proof. We have identified the two spaces pE{kerTqn andEn{pkerT
qn isometrically for eachnPN. Hence the diagram (1.3.7) implies thatT is a multi-isometry if and only if Tpnqis an isometry for eachnPN, and, by Proposition 1.4(i), the latter happens if and only ifTpnqis a quotient operator for each nP
N.
Corollary 3.8. Let F be a closed subspace of a power-normed spaceE. Then the iso-
morphism J1
F : E1{FK Ñ F1 induced by the dual of the inclusion JF : F Ñ E is a
Proof. TakenPN. By Proposition 3.7, we must show thatpJF1 qp nqis a quotient operator. SincepJ1 Fqp nq“ pJpnq F q1 andJ pnq
F is an isometry, this follows from Proposition 1.4(ii).
Proposition 3.9. Let E be an infinite-dimensional Banach space. Then:
(i) for 1 ď pă 8, the power-normed spaces pEn, µp,nq and pEn,} ¨ }`p
npEqq are not
multi-isomorphic;
(ii) for1 ăpď 8, the power-normed spacespEn, } ¨ }`p
npEqq and pE
n, ν
p,nq are not multi-isomorphic.
Proof. (i) Assume towards a contradiction that there is a multi-isomorphismT PBpEq, whereTpnqmapspEn, µ
p,nqontopEn, }¨}`p
npEqqfor eachnPN. Then we have
sup !› › ›T pnq:pEn, µp,nq Ñ pEn, } ¨ }`p npEqq › › ›:nPN ) ă 8.
By Example 3.3 (in the case wherep“q), this means thatT is ap–summing operator, which contradicts the fact thatT is an isomorphism on an infinite-dimensional Banach space. Indeed, the composition of any two p–summing operators is compact (see [24, p. 50]), and hence the isomorphismT2would be compact if T werep–summing.
(ii) This follows easily by duality.
Note that, for eachnPN, we have the equalitiesµ8,n“ } ¨ }`8
npEqby equation (1.5.1)
and} ¨ }`1
npEq“ν1,nby equation (1.5.9).
Corollary 3.10. Let E be an infinite-dimensional Banach space, and take p with
1 ďp ď 8. Then the p–multi-normed spaces pEn, µp,nq and pEn, νp,nq are not multi-
isomorphic.
Proof. For 1 ďp ă 8, this follows immediately by combining Proposition 3.9(i) with the inequality (1.5.12), while the case where p“ 8 follows from equation (1.5.1) and Proposition 3.9(ii).
There is a quantitative version of Proposition 3.9 and Corollary 3.10 in the case where E is a finite-dimensional space. Indeed, suppose that dimE“k. Then
?
kď }IE :pEn, µp,nq Ñ pEn, νp,nq}mbďk .
The upper bound follows from equation (1.4.4). The lower bound follows from equation (3.1.1) in the case where 1ďp ď2 because πppEq ě π2pEq “
?
k; in the case where 2ďpď 8, it follows by duality. It can be shown that both these bounds are optimal to within a multiplicative constant.
Example3.11. We shall show that, for eachpwith 1ďpď 8, the inverse of a bijective multi-contraction need not be multi-bounded, and hence there is no analogue of the Banach isomorphism theorem for multi-bounded operators.
(We remark that, in the setting of multi-norms themselves, several examples of the failure of the Banach isomorphism theorem were given in [20]. For example, Example 6.25 of [20] shows that there are multi-norms based on infinite-dimensional Banach spacesE and F such that MpE, Fq “BpE, Fq, but MpF, Eq “ NpF, Eq, the nuclear operators
from F toE, and Example 6.30 of [20] shows that the analogue of the Banach isomor- phism theorem may fail even when there is one multi-norm based on a Banach spaceE and we consider operators inBpEq. See [20, Example 6.39] for a further example.)
In the present situation, take pwith 1 ďp ď 8and take any infinite-dimensional Banach space E, and consider the identity operatorIE on E. Equation (1.5.13) shows that › › ›I pnq E :pE n, νp,nq Ñ pEn, µ p,nq › › ›ď1 pnPNq,
but Corollary 3.10 implies that its inverse is not multi-bounded, so that
› › ›I pnq E :pE n, µp,nq Ñ pEn, ν p,nq › › ›Ñ 8 as nÑ 8.
With a little more work, we can present a similar example forstrongp–multi-normed spaces. In the case where eitherp“2 orp“ 8, this follows immediately from the above example by Theorem 2.25. Otherwise, takeE “`p. By Example 2.27, (ii) and (iii), the p–sum normp} ¨ }`p
npEqqand the weakp–summing normpµp,nqare strongp–multi-norms based onE. Equation (1.5.2) shows that
› › ›I pnq E :pE n, } ¨ }`p npEqq Ñ pE n, µp,nq›› ›ď1 pnPNq,
but, by Proposition 3.9(i), its inverse is not multi-bounded, so that
› › ›I pnq E :pE n, µp,nq Ñ pEn, } ¨ }`p npEqq › › ›Ñ 8 as nÑ 8.
This provides the required example.
Example 3.12. Let F be a closed subspace of a Banach space E, and take p with
1ďpď 8. Equation (1.5.8) shows that the inclusionJF :F ÑEis a multi-isometry with respect to the minimump–multi-normspµFp,nqandpµEp,nqbased onF andE, respectively. In contrast, suppose thatF andE are endowed with their maximump–multi-norms pνp,nF qand pνp,nE q, respectively. Proposition 1.19 implies that JF is a multi-contraction, but it is not always a multi-embedding. Indeed, suppose that 1ăpă 8andp‰2 and that E and F have been chosen as in Example 1.30. Then equation (1.6.3) shows that JF is not a multi-embedding ofpFn, νF
p,nqinto pEn, νp,nE q.
Example3.13. Again, letF be a closed subspace of a Banach spaceE, and takepwith 1ďpď 8. We observe that, by Propositions 1.20 and 3.7,νp,nE{F is equal to the quotient norm onpE{Fqn of the normνE
p,n onEn for eachnPN.
However, the analogous result may fail for the minimump–multi-norm. To see this, take q with 1 ă q ă 8 and q ‰ 2, and choose E and F as in Example 1.30. Then it follows from equation (1.6.4) that theq–multi-normed space ppE1{FKqn, µE1{FK
q,n qis not multi-isomorphic to theq–multi-normed spaceppE1{FKqn, µE1
q,nq, whereµE 1
q,n denotes the quotient norm onpE1{FKqn of the normµE1
q,n onpE1q n forn
PN.
We have noted in Theorem 2.18 that multi-norms correspond toc0–norms onc0bE.
Suppose thatpEn,
} ¨ }nqand pFn,} ¨ }nqare multi-normed spaces. ThenT PBpE, Fq is multi-bounded if and only ifIc0bT is bounded as a map from c0bE toc0bF, and
as the ‘op´erateurs r´eguliers’ of [45, D´efinition 3.2] (where they are defined in the special case that E and F are Banach lattices). More generally, take p with 1 ď pă 8 and suppose thatpEn,
} ¨ }nqandpFn,
} ¨ }nqarep–multi-normed spaces. Thenp–multi-norms based on E correspond to `p–norms on `pbE, where the correspondence is given in equation (2.4.3). Thus the following theorem follows from Theorem 2.21.
Theorem 3.14. Take p with 1 ď p ă 8, and suppose that pEn,
} ¨ }nq and pFn, } ¨ }nq
arep–multi-normed spaces. TakeT PBpE, Fq. Then T isp–multi-bounded if and only if
I`pbT is bounded as a map from`pbE to`pbF; in this case,}T}mb“ }I`pbT}.
Let tE0, E1u and tF0, F1u be two compatible couples of complex Banach spaces,