Abstract
We study, among others, upper, lower, upper modified and lower modified n-th von Neumann–Jordan constant and relationships between them. There are characterized uniformly non-\(l_{n}^{1}\) Banach spaces in terms of the upper modified n-th von Neumann–Jordan constant. Moreover, this constant is calculated explicitly for Lebesgue spaces \(L^{p}\) and \(l^{p}\)\((1\le p\le \infty ).\) Finally, it is shown that the sequence of n-th upper and modified upper von Neumann–Jordan constants for the space \(L^p\) as well as \(l^p\)\((2<p<\infty )\) converges to \(B_p^2\), where \(B_p\) is the best type (2, p) constant in the Khinthine inequality for the case \(2\le p<\infty \).
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1 Introduction
Let S(X) (resp. B(X)) be the unit sphere (resp. the unit closed ball) of a real Banach space \((X,\left\| \cdot \right\| _{X}).\) The letters \( {\mathbb {Z}},\)\({\mathbb {N}}\) and \({\mathbb {R}}\) stand for the sets of integers, positive integers and real numbers, respectively. For any subset \(A\subset X, \) denote \(A^{n}=\underset{n\text {-times}}{\underbrace{A\times \cdots \times A} }.\) Let \((\varOmega ,\varSigma , \mu )\) be a measure space with a \(\sigma \)-finite, non-atomic and complete measure \( \mu .\) Denote by \(L^{p}(\mu )\)\((1\le p\le \infty )\) the Lebesgue space of real \(\varSigma \)-measurable functions defined on \(\varOmega .\) The symbol \( l_{m}^{p}\)\((1\le p\le \infty ,\)\(m\in {\mathbb {N}}\cup \{\infty \})\) stands for m-dimensional Lebesgue sequence space. Clearly, \(l_{\infty }^{p}=l^{p}.\)
In 1937 Clarkson [5], on the basis of the famous paper [13] by Jordan and von Neumann, introduced the constant \(C_{NJ}(X)\) (called the von Neumann–Jordan constant or NJ-constant for short) as the smallest constant \(C\ge 1\) such that
holds for any \(x,y\in X\) with \(\left\| x\right\| _{X}^{2}+\left\| y\right\| _{X}^{2}>0.\) An equivalent and more convenient definition of NJ-constant is given in [15] by the formula
The classical Jordan and von Neumann results [13] state that \(1\le C_{NJ}(X)\le 2\) for any Banach space X and \(C_{NJ}(X)=1\) if and only if X is a Hilbert space. Clarkson [5] showed that if \(1\le p\le \infty \) and \(\dim L^{p}(\mu )\ge 2,\) then \(C_{NJ}(L^{p}(\mu ))=2^{2/\min \{p,q\}-1}, \) where \(1/p+1/q=1.\) Kato and Takahashi [16], observed that \(C_{NJ}(X)=C_{NJ}(X^{*}).\) Moreover, they proved that if the Banach space X is uniformly convex, then \(C_{NJ}(X)<2\) and if \(C_{NJ}(X)<2,\) then X admits an equivalent uniformly convex norm. The same authors [16] state that the Banach space X is uniformly non-square if and only if \( C_{NJ}(X)<2.\) Some results concerning relationships between von Neumann–Jordan and so called James constant have been obtained among others in [2, 15, 19, 21, 23, 25].
A similar constant
was introduced in 2006 by Gao [7] and called the modified von Neumann–Jordan constant. It is clear that \(C_{NJ}^{^{\prime }}(X)\le C_{NJ}(X).\) It has been shown that \(C_{NJ}^{^{\prime }}(X)\) does not necessarily coincide with \(C_{NJ}(X)\) (see [2, 8]). These constants have been considered recently also in [22].
The von Neumann Jordan constant has been generalized in many directions (see e.g. [17, 24, 26]).
To generalize the von Neumann–Jordan constant, denote
for any \(x_{1},x_{2},\ldots ,x_{n}\in X\) such that \(\sum _{j=1}^{n}\left\| x_{j}\right\| _{X}^{2}>0.\)
Definition 1
The smallest, resp., the largest constant \(C>0\) such that
for all \(x_{j}\in X,\)\((j=1,2,\ldots ,n\) and \(n\ge 2)\) with \( \sum _{j=1}^{n}\left\| x_{j}\right\| _{X}^{2}>0\) is called an upper, resp., lowern-th von Neumann–Jordan constant and denoted by \({\overline{C}}_{NJ}^{(n)}(X)\), resp., \({\underline{C}} _{NJ}^{(n)}(X).\) If the infimum, resp., supremum of C satisfying (1) is taken over all \(x_{j}\in S(X),\)\((j=1,2,\ldots ,n\) and \(n\ge 2),\) then it is called upper, resp.,lowermodifiedn- th von Neumann–Jordan constant and denoted by \({\overline{C}} _{mNJ}^{(n)}(X),resp.,\)\({\underline{C}}_{mNJ}^{(n)}(X).\)
It is well known that \({\overline{C}}_{NJ}^{(2)}(X)=\left[ {\underline{C}} _{NJ}^{(2)}(X)\right] ^{-1}=C_{NJ}(X)\) (see [20]). As it is proved below, the equality \({\overline{C}}_{NJ}^{(n)}(X)=\left[ {\underline{C}} _{NJ}^{(n)}(X)\right] ^{-1}\) is not true in general for \(n>2.\) Moreover, \( {\overline{C}}_{mNJ}^{(2)}(X)=C_{NJ}^{\prime }(X)\) and \({\overline{C}} _{mNJ}^{(n)}(X)\) need not be equal to \(\left[ {\underline{C}}_{mNJ}^{(n)}(X) \right] ^{-1}\) even for \(n=2\) (see [7]). The n-th von Neumann–Jordan constant introduced and investigated by Kato, Takahashi and Hashimoto in [17] is exactly the upper n-th von Neumann–Jordan constant.
In 1964 James [12] introduced the notion of uniformly non-\(l_{n}^{1}\) Banach space. Namely, a Banach space X is called uniformly non-\( l_{n}^{1}\) if there exists \(\delta >0\) such that for each n elements of the unit ball B(X)
(see [10]). The definition remains the same if we replace the unit ball B(X) by the unit sphere S(X). If X is uniformly non-\(l_{n}^{1}\) for \( n=2,\) then it is called uniformly non-square. It is worth mentioning that uniform non-squareness plays a crucial role in fixed point theory, since any uniformly non-square Banach space has the fixed point property (for more details see [9]). In 1987 Kamińska and Turett [14] proved that the uniform non-\(l_{n}^{1}\) for Banach spaces is equivalent to the fact that there exists \(\delta >0\) such that for all \(x_{1},x_{2},\ldots ,x_{n}\) in X
Banach spaces that are uniformly non-\(l_{n}^{1}\) for a certain \(n\in {\mathbb {N}}\) have been studied by A. Beck [3]. Such spaces are said to be B- convex. Beck [3] proved that a Banach space X is B-convex if and only if a certain strong law of large numbers is valid for random variables with ranges in X. Moreover, B-convexity is a very important property in fixed point theory because every B-convex uniformly monotone Köthe space has the fixed point property (see [1]).
2 Basic Properties
Proposition 1
Let \(n\ge 2\) and X be a Banach space. The lower, upper, modified lower and modified upper n-th von Neumann–Jordan constants have the following properties:
-
(a)
\(1\le {\overline{C}}_{mNJ}^{(n)}(X)\le {\overline{C}} _{NJ}^{(n)}(X)\le n\) and \(1/n\le {\underline{C}}_{NJ}^{(n)}(X)\le {\underline{C}}_{mNJ}^{(n)}(X)\le 1;\)
-
(b)
\({\overline{C}}_{NJ}^{(n)}(X)\le {\overline{C}} _{NJ}^{(n+1)}(X)\) and \({\underline{C}}_{NJ}^{(n+1)}(X)\le {\underline{C}} _{NJ}^{(n)}(X);\)
-
(c)
\({\overline{C}}_{mNJ}^{(n)}(X)\le \frac{n+1}{n}{\overline{C}}_{mNJ}^{(n+1)}(X).\)
Proof
Let \((X,\left\| \cdot \right\| _{X})\) be a Banach space and \(n\ge 2.\)
(a) The estimation \({\overline{C}}_{NJ}^{(n)}(X)\le n\) is proved in [17]. \({\overline{C}}_{mNJ}^{(n)}(X)\le {\overline{C}}_{NJ}^{(n)}(X)\) by the definition. Putting \(x_{1}\in S(X)\) and \(x_{i}=x_{1}\) for \(i=2,3,\ldots ,n,\) we have
Hence
and
Obviously, \({\underline{C}}_{NJ}^{(n)}(X)\le {\underline{C}}_{mNJ}^{(n)}(X)\) by the definition. To prove that \(\frac{1}{n}\le {\underline{C}}_{NJ}^{(n)}(X),\) we use the mathematical induction principle. For \(n=2\) we have
Suppose that \({\underline{C}}_{NJ}^{(n-1)}(X)\ge \frac{1}{n-1}.\) Notice that
for any \(x,y\in X.\) Really, since
we have
Hence
for any \(i=1,2,\ldots ,n.\) It follows that
and consequently \({\underline{C}}_{NJ}^{(n)}(X)\ge \frac{1}{n},\) which finishes the proof of (a).
(b) The inequality \({\overline{C}}_{NJ}^{(n)}(X)\le {\overline{C}} _{NJ}^{(n+1)}(X)\) is proved in [17]. To prove the second inequality it is enough to notice that
for any elements \(x_{1},x_{2},\ldots ,x_{n}\in X.\) Hence
(c) For any \(x_{1},x_{2},\ldots ,x_{n+1}\in S(X),\) by the inequality (2), we have
whence \({\overline{C}}_{mNJ}^{(n)}(X)\le \frac{n+1}{n}{\overline{C}} _{mNJ}^{(n+1)}(X).\)
\(\square \)
Proposition 2
Let \(n\ge 2\) and X be a Banach space.
-
(a)
The following conditions are equivalent:
-
(i)
X is a Hilbert space;
-
(ii)
\({\overline{C}}_{NJ}^{(n)}(X)=1;\)
-
(iii)
\({\underline{C}}_{NJ}^{(n)}(X)=1.\)
-
(i)
-
(b)
If X is a Hilbert space, then \({\overline{C}} _{mNJ}^{(n)}(X)={\underline{C}}_{mNJ}^{(n)}(X)=1.\)
Proof
(a) By Theorem 5 (iii) in [17], conditions (i) and (ii) are equivalent. To prove the implication \(\mathrm{(i)}\Rightarrow \mathrm{(iii)}\) suppose that X is a Hilbert space. By elementary calculations, we get that
for any elements \(x_{1},x_{2},\ldots ,x_{n}\in X,\) whence \({\underline{C}} _{NJ}^{(n)}(X)=1.\) Conversely, if \({\underline{C}}_{NJ}^{(n)}(X)=1,\) then, by Proposition 1 (a) and (b), we have
Hence \(C_{NJ}(X)=1.\) Consequently, X is a Hilbert space (see [13]).
(b) follows immediately from (3). \(\square \)
In general, the explicit calculation of various types of n-th von Neumann–Jordan constant is rather a hard problem. Anyway, the next proposition can be helpful to do this.
Proposition 3
Let \((X,\left\| \cdot \right\| _{X})\) be a Banach space and \(n\ge 2.\) Denote \(D_{1}=\left[ B(X)\right] ^{n}\setminus \{ {\mathbf {0}}\},\)\(D_{2}=B(l_{n}^{2}(X))\setminus \{{\mathbf {0}}\},\)\( D_{3}=S(l_{n}^{2}(X)),\) where \({\mathbf {0}}=(0,0,\ldots ,0).\) Then
and
for any \(j=1,2,3.\)
Proof
Since
it follows that
where \({\mathbf {x}}=(x_{1},x_{2},\ldots ,x_{n}).\) To show (4), it remains to prove that \(\sup _{{\mathbf {x}}\in D_{3}}C^{(n)}({\mathbf {x}})\ge {\overline{C}} _{NJ}^{(n)}(X).\) Let \({\mathbf {x}}=\left( x_{1},x_{2},\ldots ,x_{n}\right) \in X^{n}\setminus \{{\mathbf {0}}\}.\) Define the sequence \({\mathbf {y}} =(y_{k})_{k=1}^{n}\) by
for \(k=1,2,\ldots ,n\) and \(n\ge 2.\) Obviously, \({\mathbf {y}}\in S(l_{n}^{2}(X)).\) Hence
for any elements \({\mathbf {x}}\in X^{n}\setminus \{{\mathbf {0}}\}.\) Therefore
which finishes the proof of (4). The equality (5) can be proved similarly.
\(\square \)
Let \(n\ge 2.\) Define
and for each integers \(n>2\)
where \({\mathbf {1}}\) denotes the \(2^{n-2}\)-by-1 column vector in which all the elements are equal to 1. The matrix \(A_{n}\) generates a linear operator
defined for any \(x\in l_{n}^{2}(X)\) by the formula
A one-to-one correspondence between n-th von Neumann–Jordan constant \( {\overline{C}}_{NJ}^{(n)}(X)\) and the norm of the operator \(T_{n}\) is given by the following result.
Corollary 1
Let \((X,\left\| \cdot \right\| _{X})\) be a Banach space and \( T_{n}:l_{n}^{2}(X)\rightarrow l_{2^{n-1}}^{2}(X)\) be the linear operator generated by the matrix \(A_{n}.\) Then
for any integer \(n\ge 2.\)
Proof
Fix \(n\ge 2.\) Let \(x_{1},x_{2},\ldots ,x_{n}\in X\) and \( \sum _{j=1}^{n}\left\| x_{j}\right\| _{X}^{2}>0.\) Denote \( {\mathbf {x}}=(x_{1},x_{2},\ldots ,x_{n}).\) By Proposition 3, we have
\(\square \)
Corollary 2
Let \(\left( X^{*},\left\| \cdot \right\| _{X^{*}}\right) \) be the dual space of the Banach space \((X,\left\| \cdot \right\| _{X}).\) Then
-
(a)
\({\underline{C}}_{NJ}^{(n)}(X^{*})\ge \frac{1}{ {\overline{C}}_{NJ}^{(n)}(X)}.\)
-
(b)
\({\underline{C}}_{NJ}^{(n)}(X)\ge \frac{1}{{\overline{C}} _{NJ}^{(n)}(X^{*})}.\)
Proof
(a) Define an operator \(T_{n}:l_{n}^{2}(X)\rightarrow l_{2^{n-1}}^{2}(X)\) as above. Let \(T_{n}^{*}\) be the adjoint of operator \(T_{n}.\) Obviously, \(T_{n}^{*}:l_{2^{n-1}}^{2}(X^{*})\rightarrow l_{n}^{2}(X^{*})\) is generated by the matrix \(A_{n}^{*}=A_{n}^{T}.\) Then, by Corollary 1, we get
for any \(y^{*}=(y_{1}^{*},y_{2}^{*},\ldots ,y_{2^{n-1}}^{*})\in l_{2^{n-1}}^{2}(X^{*})\setminus \{{\mathbf {0}}\}.\) Let \(S_{n}^{*}= \frac{1}{2^{n-1}}A_{n}.\) Then, \(S_{n}:l_{n}^{2}(X^{*})\rightarrow l_{2^{n-1}}^{2}(X^{*}).\) Since
where \(I_{n}\) is the identity matrix of size \(n\times {n},\) it follows that
for any \(x^{*}=(x_{1}^{*},x_{2}^{*},\ldots ,x_{n}^{*})\in l_{n}^{2}\left( X^{*}\right) .\) Hence, by (6), we have
for any \((x_{1}^{*},x_{2}^{*},\ldots ,x_{n}^{*})\in \left( X^{*}\right) ^{n}\setminus \{{\mathbf {0}}\}.\) By the definition of the lower n-th von Neumann–Jordan constant, we get the thesis of (a).
(b) Since X can be isometrically embedded into \(X^{**},\) it follows that \({\underline{C}}_{NJ}^{(n)}(X)\ge {\underline{C}} _{NJ}^{(n)}(X^{**}).\) Hence, by (a), we have
\(\square \)
The n-th von Neumann–Jordan constant for some classical Banach spaces can be calculated effectively.
Proposition 4
Let \(n\ge 2.\)
-
(a)
If \(1\le p\le 2\) and \(n\le m\le \infty ,\) then \( {\overline{C}}_{NJ}^{(n)}(l_{m}^{p})={\overline{C}}_{mNJ}^{(n)}(l_{m}^{p})=n^{ \frac{2}{p}-1}.\)
-
(b)
If \(2^{n-1}\le m\le \infty ,\) then \({\overline{C}} _{NJ}^{(n)}(l_{m}^{\infty })={\overline{C}}_{mNJ}^{(n)}(l_{m}^{\infty })=n.\)
-
(c)
If \(n\le m\le \infty ,\) then \({\underline{C}} _{NJ}^{(n)}(l_{m}^{\infty })={\underline{C}}_{mNJ}^{(n)}(l_{m}^{\infty })= \frac{1}{n}.\)
-
(d)
Let \((\varOmega ,\varSigma ,\mu )\) be a measure space with non-atomic \(\sigma \)-finite and complete measure \(\mu .\) Then
$$\begin{aligned} {\overline{C}}_{mNJ}^{(n)}(L^{1}(\mu ))={\overline{C}}_{NJ}^{(n)}(L^{1}(\mu ))= {\overline{C}}_{NJ}^{(n)}(L^{\infty }(\mu ))={\overline{C}}_{NJ}^{(n)}(L^{\infty }(\mu ))=n \end{aligned}$$and
$$\begin{aligned} {\underline{C}}_{mNJ}^{(n)}(L^{\infty }(\mu ))={\underline{C}} _{NJ}^{(n)}(L^{\infty }(\mu ))=\frac{1}{n}. \end{aligned}$$
Proof
(a) Let \(n\le m\le \infty .\) By Theorem 3 (ii) in [17], \( {\overline{C}}_{NJ}^{(n)}(l_{m}^{p})=n^{\frac{2}{p}-1}.\) By Proposition 1 (b), \({\overline{C}}_{mNJ}^{(n)}(l_{m}^{p})\le n^{\frac{2 }{p}-1}.\) Taking the canonical basis \(\left( e_{i}\right) _{i=1}^{m}\)\( (n\le m\le \infty )\) in \(l_{m}^{p},\) we get
Since \(e_{i}\in S(l_{m}^{p})\) for \(i\in {\mathbb {N}}\cap [1,m],\) it follows that
Hence \({\overline{C}}_{mNJ}^{(n)}(l_{m}^{p})=n^{\frac{2}{p}-1}.\)
(b) Let \(2^{n-1}\le m\le \infty .\) Then \(l_{m}^{\infty }\) is not uniformly non-\(l_{n}^{1}.\) By Theorem 5 (iv) in [17] and Proposition 1 (a), we conclude that \({\overline{C}} _{NJ}^{(n)}(l_{m}^{\infty })=n.\) To prove that \({\overline{C}}_{mNJ}^{(n)}(l_{m}^{\infty })=n\) take the matrix \(A_{n}\) defined as above. The column j of \( A_{n}\) denote by \({\mathbf {a}}_{j}^{(n)}.\) Then \({\mathbf {a}} _{j}^{(n)}=[1,a_{2j}^{(n)},\ldots ,a_{2^{n-1}j}^{(n)}],\) where \(a_{ij}^{(n)}=\pm 1\) for any \(1\le j\le n\) and \(2\le i\le 2^{n-1}.\) Define for \(j=1,2,\dots ,n\),
where \(\left( e_{i}\right) _{i=1}^{m}\)\((n\le m\le \infty )\) is the canonical basis in \(l_{m}^{\infty }.\) Obviously, \(\left\| z_{j}\right\| _{l_{m}^{\infty }}=1.\) Let \((1,\theta _{2},\ldots ,\theta _{n})\) be an arbitrary sequence such that \(\theta _{j}=\pm 1\) for any \(2\le j\le n.\) Then there is exactly one row \(i_{0}\) of \(A_{n}\) such that
Hence
Moreover
for any \(i\not =i_{0}.\) Consequently,
whence
Therefore
which completes the proof of (b).
(c) Let \(n\le m\le \infty \) and \((e_{i})\) be the canonical basis in \(l_{m}^{\infty }.\) By Proposition 1 (a), we have
(d) Since \(L^{1}(\mu )\) contains an isometric copy of \(l^{1},\) applying Proposition 4 (a) for \(p=1,\) we get
Hence \({\overline{C}}_{mNJ}^{(n)}(L^{1}(\mu ))={\overline{C}}_{NJ}^{(n)}(L^{1}( \mu ))=n.\) Using the same arguments, by Proposition 4 (b) we conclude that \({\overline{C}}_{NJ}^{(n)}(L^{\infty }(\mu ))={\overline{C}} _{mNJ}^{(n)}(L^{\infty }(\mu ))=n.\) Similarly, by Proposition 4 (c), we obtain
which completes the proof. \(\square \)
3 Uniformly non-\(l_{n}^{1}\) spaces
The next theorem gives some characterizations of the uniform non-\(l_{n}^{1}\) property for Banach spaces. Kato, Takahashi and Hashimito proved in [17] that \((X,\left\| \cdot \right\| _{X})\) is uniformly non-\( l_{n}^{1}\) iff \({\overline{C}}_{NJ}^{(n)}(X)<n.\) We will extend their results.
Theorem 1
Let \((X,\left\| \cdot \right\| _{X})\) be a Banach space. Then the following conditions are equivalent:
-
(a)
\({\overline{C}}_{mNJ}^{(n)}(X)<n;\)
-
(b)
\((X,\left\| \cdot \right\| _{X})\) is uniformly non-\(l_{n}^{1};\)
-
(c)
There exists \(\delta \in (0,1)\) such that for any element \((x_{1},x_{2},\ldots ,x_{n})\in B\left( l_{n}^{2}\left( X\right) \right) ,\) we have
$$\begin{aligned} \min _{\theta _{j}=\pm 1}\left\| x_{1}+\sum \nolimits _{j=2}^{n}\theta _{j}x_{j}\right\| _{X}\le \sqrt{n}(1-\delta ); \end{aligned}$$(7) -
(d)
There exists \(\delta \in (0,1)\) such that for any element \((x_{1},x_{2},\ldots ,x_{n})\in S\left( l_{n}^{2}\left( X\right) \right) ,\) the inequality (7) is satisfied.
Proof
\(\mathrm{(a)}\Rightarrow \mathrm{(b)}.\) Suppose that \({\overline{C}} _{mNJ}^{(n)}(X)<n.\) Then
for any \(x_{1},x_{2},\ldots ,x_{n}\in S\left( X\right) .\) Since on the left hand side we have an arithmetic mean, there is at least one sequence \((1, {\overline{\theta }}_{2},\ldots ,{\overline{\theta }}_{n})\) such that
Hence
where \(\delta =\frac{\sqrt{n}-\sqrt{{\overline{C}}_{mNJ}^{(n)}(X)}}{\sqrt{n}}.\) Consequently, \(\left( X,\left\| \cdot \right\| _{X}\right) \) is uniformly non-\(l_{n}^{1}.\)
\(\mathrm{(b)}\Rightarrow \mathrm{(c)}.\) Assume that \(\left( X,\left\| \cdot \right\| _{X}\right) \) is uniformly non-\(l_{n}^{1}.\) Let \((x_{1},x_{2},\ldots ,x_{n})\in B\left( l_{n}^{2}\left( X\right) \right) .\) Since \(\sum _{j=1}^{n}\left\| x_{j}\right\| _{X}^{2}\le 1,\) it follows that \(\min _{1\le j\le n}\left\| x_{j}\right\| _{X}\le \frac{1}{\sqrt{n}}.\) Moreover, by the Hölder inequality, we have
Case 1. Suppose that \(\frac{1}{2\sqrt{n}}<\min _{1\le j\le n}\left\| x_{j}\right\| _{X}\le \frac{1}{\sqrt{n}}.\) By the characterization of uniform non-\(l_{n}^{1}\) given in [14] and by the inequality (8), there is \(\delta _{1}>0\) such that
Case 2. Suppose that \(0\le \min _{1\le j\le n}\left\| x_{j}\right\| _{X}\le \frac{1}{2\sqrt{n}}.\) Let \(x_{k}\) be the element on which the minimum is taken. Then, by the Hölder inequality, we have
for any choice of signs. Define
for any \(t\in \left[ 0,\frac{1}{2\sqrt{n}}\right] .\) By elementary calculus, we conclude that f is an increasing function on the interval \(\left[ 0, \frac{1}{2\sqrt{n}}\right] .\) Hence, the function f(t) takes its highest value on \(\left[ 0,\frac{1}{2\sqrt{n}}\right] \) at the point \(t=\frac{1}{2 \sqrt{n}}.\) Thus,
for any choice of signs. Taking
we get (c).
\(\mathrm{(c)}\Rightarrow \mathrm{(d)}.\) It is obvious.
\(\mathrm{(d)}\Rightarrow \mathrm{(a)}.\) Let \((x_{1},x_{2},\ldots ,x_{n})\in S\left( l_{n}^{2}\left( X\right) \right) .\) By the assumption (d) there exists \( \delta \in \left( 0,1\right) \) such that
for some choice of signs. Moreover, by (8), \(\left\| x_{1}\pm x_{2}\pm \cdots \pm x_{n}\right\| _{X}\le \sqrt{n}\) for any choice of signs. Hence, we have
By the definition of the upper n-th von Neumann–Jordan constant \({\overline{C}}_{NJ}^{(n)}(X)\) and Proposition 3, we conclude
which finishes the proof. \(\square \)
By Theorem 1 and the definition of B-convexity, we get immediately
Corollary 3
A Banach space \((X,\left\| \cdot \right\| _{X})\) is B-convex if and only if there is \(n\ge 2\)\((n\in N)\) such that \({\overline{C}} _{mNJ}^{(n)}(X)<n.\)
Notice that \({\overline{C}}_{mNJ}^{(n)}(X)\) is not equal to \({\overline{C}} _{NJ}^{(n)}(X)\) in general (for \(n=2\) see [18]).
Corollary 4
\({\overline{C}}_{mNJ}^{(n)}(X)=n\) if and only if \({\overline{C}} _{NJ}^{(n)}(X)=n.\)
Proof
Since \((X,\left\| \cdot \right\| _{X})\) is uniformly non-\(l_{n}^{1}\) iff \({\overline{C}}_{NJ}^{(n)}(X)<n,\) it follows, by Theorem 1, that \({\overline{C}}_{NJ}^{(n)}(X)<n\) iff \({\overline{C}} _{mNJ}^{(n)}(X)<n.\) Hence, by Proposition 1 (a), we get the thesis. \(\square \)
Remark 1
Let us notice that the above corollary can be reformulated equivalently as follows
4 Upper and lower n-th von Neumann–Jordan constant for \(L^{p}\)-spaces
Now we will calculate the upper n-th von Neumann–Jordan constant for Lebesgue spaces \(L^{p}(\mu )\) and \(l_{m}^{p}\)\((1<p<\infty \), \(1\le m\le \infty ).\) To prove the next lemma, we will apply the following results given by Figiel, Iwaniec and Pełczyński in [6]. Namely, for arbitrary scalars \( c_{1},c_{2},\ldots ,c_{n}\) and \(2<p<\infty \) we have
where \(r_{1},r_{2},\ldots ,r_{n}\)\((n=1,2,\ldots )\) are Rademacher functions, that is \(r_{n}(t)=\) sign\(\left( \sin 2^{n}\pi t\right) .\)
Let \(\left\lfloor \cdot \right\rfloor :{\mathbb {R}}\rightarrow {\mathbb {Z}}\) be the floor function, i.e. \(\left\lfloor x\right\rfloor =\max \left\{ k\in {\mathbb {Z}}:k\le x\right\} \) for any \(x\in {\mathbb {R}}.\)
Lemma 1
Let \(2<p<\infty \) and \(X=L^{p}(\mu )\) or \(X=l^{p}.\) Then
for any \(x_{1},x_{2},\ldots ,x_{n}\in X\) and any integer \(n\ge 1.\)
Proof
Fix an integer \(n\ge 1.\) Notice that
for any scalars \(c_{1},c_{2},\ldots ,c_{n}.\) On the other hand, it can be proved elementarily that
Suppose that \(x_{k}=\left( t_{i}^{(k)}\right) _{i=1}^{\infty }\in l^{p}\) for \(k=1,2,\ldots ,n.\) By the inequality (9), we get
for any \(i\in {\mathbb {N}}.\) Summing by sides from \(i=1\) to \(\infty \) and reversing the order of summation, we obtain the thesis. Similarly, for \( X=L^{p}(\mu )\) take \(x_{1},x_{2},\ldots ,x_{n}\in L^{p}(\mu ).\) Then, by the inequality (9), we get
for almost every \(t\in \varOmega .\) Integrating by sides this inequality and reversing the order of summation and integration, we obtain the desired inequality. \(\square \)
Theorem 2
Let \(1\le p<\infty \) and \(X=L^{p}(\mu )\) or \(X=l_{m}^{p}\)\((1\le m\le \infty ).\) Then
Proof
Case 1. Let \(1\le p\le 2.\) By Theorem 3 from [17] and Proposition 1 (a), for all \(n\ge 2,\) we have
whenever \(X=L^{p}(\mu )\) or \(X=l_{m}^{p},\)\(m\in {\mathbb {N}}\cup \{\infty \}.\) The opposite inequality follows immediately from Proposition 4 (a) whenever \(X=l_{m}^{p}\) with \(m\ge n.\)
Now consider \(X=L^{p}(\mu ).\) Let \(A\subset \varOmega \) be a set of positive finite measure. Divide the set A into n pairwise disjoint subsets \( A_{1}, \)\(A_{2},\ldots ,A_{n}\) such that \(\bigcup \limits _{i=1}^{n}A_{i}=A\) and \( \mu (A_{i})=\frac{1}{n}\mu (A).\) Define \(z_{i}=\mu (A_{i})^{-1/p}\chi _{A_{i}}\) for \(i=1,2,\ldots ,n.\) Then
for any \(i\in \{1,2,\ldots ,n\}\) and
Hence \({\overline{C}}_{mNJ}^{(n)}(L^{p}(\mu ))={\overline{C}}_{NJ}^{(n)}(L^{p}(\mu ))=n^{ \frac{2}{p}-1},\) whenever \(1\le p\le 2.\)
Case 2. Let \(2<p<\infty \) and \(X=L^{p}(\mu )\) or \(X=l_{m}^{p}\)\((2^{n-1}\le m\le \infty )\). By the Hölder-Rogers inequality for \(p>2\) and by Lemma 1, we have
for any \(x_{1},x_{2},\ldots ,x_{n}\in X.\) Assume that \(x_{1},x_{2},\ldots ,x_{n}\in S(X).\) Then, by inequality (10), we have
whence
Let the matrix \(A_{n}\) be defined as in the proof of Proposition 3 (b). Denote by \(y_{i}\) column i of the matrix \(A_{n}\)\((i=1,2,\ldots ,n).\) For any \(i\in \left\{ 1,2,\ldots ,n\right\} \) define
where \(y_{i}^{T}\) denotes the transpose of the column \(y_{i}.\) Then
for any \(i\in \left\{ 1,2,\ldots ,n\right\} .\) Hence \(z_{1},z_{2},\ldots ,z_{n}\in S\left( l_{2^{n-1}}^{p}\right) .\) For any element \( x=(t_{1},t_{2},\ldots ,t_{2^{n-1}})\in l_{2^{n-1}}^{p}\) denote by \(x^{*}\) its non-increasing rearrangement, i.e. a non-increasing sequence obtained from \(\left\{ \left| t_{i}\right| \right\} _{i=1}^{2^{n-1}}\) by a suitable permutation of the integers. Notice that for all sequences \( (1,\theta _{2},\ldots ,\theta _{n})\) such that \(\theta _{j}=\pm 1,\)\( (j=2,3,\ldots ,n)\) the non-increasing rearrangements \(\left( z_{1}+\sum _{j=2}^{n}\theta _{j}z_{j}\right) ^{*}\) coincide. Denoting by \( \left( v_{1},v_{2},\ldots ,v_{2^{n-1}}\right) \) the non-increasing sequence such that
for any sequences \((1,\theta _{2},\ldots ,\theta _{n}).\) Hence
Notice that \(v_{1}=\frac{n}{\left( 2^{n-1}\right) ^{1/p}},\)\(v_{l}=\frac{n-2k }{\left( 2^{n-1}\right) ^{1/p}}\) for \(\left( {\begin{array}{c}n\\ k\end{array}}\right) \) subsequent integers l, \(\left( k=1,2,\ldots ,\left\lfloor n/2\right\rfloor \right) .\) Consequently,
Since \(l_{2^{n-1}}^{p}\) can be embedded isometrically in any \(l_{m}^{p}\) with \( m\ge 2^{n-1},\) by inequality (11) applied for \(X=l_{m}^{p}\), it follows that
whenever \(2<p\le \infty \) and \(m\ge 2^{n-1}.\)
Since \(L^{p}(\mu )\) contains an isometric copy of \(l_{2^{n-1}}^{p},\) by inequality (11), we obtain the thesis for \(X=L^{p}(\mu ),\) which completes the proof. \(\square \)
Haagerup [11] proved that the best type (2, p) constant in the Khinthine inequality for \(2\le p<\infty \) is \(B_{p}=\sqrt{2}\left( \frac{1}{\sqrt{\pi }}\varGamma \left( \frac{p+1}{2} \right) \right) ^{\frac{1}{p}}.\) Kato, Takahashi and Hashimoto proved in [17] that \({\overline{C}} _{NJ}^{(n)}\left( X\right) \le \min \left\{ n^{\frac{2}{q} -1},B_{p}^{2}\right\} .\) Combining this result with Theorem 2, we get two hand side estimation of upper von Neumann–Jordan constant for Lebesgue spaces with \(p\in (2,\infty ).\)
Corollary 5
Let \(2<p<\infty ,\)q be conjugate to p and \(X=L^{p}(\mu )\) or \(X=l_{m}^{p}.\) If \(m\ge 2^{n-1},\) then
Proof
The left hand side inequality follows immediately from Theorem 2. Namely,
whenever \(2<p<\infty ,\)\(X=L^{p}(\mu )\) or \(X=l_{m}^{p}\) and \(m\ge 2^{n-1}. \) The right hand side inequality was proved in [17]. \(\square \)
Theorem 3
Let \(2<p<\infty .\) If \(X=L^{p}(\mu )\) or \(X=l^{p},\) then
Proof
Assume that \(2<p<\infty .\) By Theorem 2, for every \(n\in {\mathbb {N}}\) , we get
where \(j=n-k.\) On the other hand, by elementary asymptotic method (see [4]), we have
for any \(x>0.\) Hence, letting \(t_{j}=j\sqrt{2/n},\) we obtain
for any \(x>0.\) Passing to the limit gives the classic de Moivre–Laplace theorem (see [4]). In our case, we get
Moreover, by the definition of the gamma function, it follows that
Thus, by the equality (12), we obtain
for any \(n\in {\mathbb {N}}\). Hence
By Proposition 1 (a), (c) and Corollary 5, we have
By the squeeze theorem, it follows that
whence \(\lim _{n\rightarrow \infty }{\overline{C}}_{mNJ}^{(n)}\left( X\right) =B_{p}^{2}.\) Furthermore, \(\lim _{n\rightarrow \infty }{\overline{C}} _{NJ}^{(n)}\left( X\right) =B_{p}^{2}\) because the sequence \(\left( {\overline{C}}_{mNJ}^{(n)}\left( X\right) \right) \) is increasing. \(\square \)
The proof of Theorem 3 presented here is a modification of idea given by Cecil Rousseau from The University of Memphis in private communication.
Corollary 6
Let \(2\le p\le \infty \) and \(X=L^{p}(\mu )\) or \(X=l_{m}^{p}.\) If \( m\ge \)n, then \({\underline{C}}_{NJ}^{(n)}(X)=n^{\frac{2}{p} -1}.\)
Proof
Let \(2\le p\le \infty \) and q denote the conjugate number of p. By Corollary 2 and Theorem 2, we have
On the other hand, taking the canonical basis \(\left\{ e_{i}\right\} _{i=1}^{n}\) in \(l_{n}^{p}\) we have
Hence, by the definition of lower n-th von Neumann–Jordan constant, we get
whenever \(m\ge n.\) Combining the both inequalities, we get the thesis. \(\square \)
Remark 2
It is known that \(C_{NJ}(X)=C_{NJ}(X^{*})\) (see [16]) and in general \({\overline{C}}_{NJ}^{(n)}(X)\ne {\overline{C}}_{NJ}^{(n)}(X^{*})\) for \(n\ge 3\) (see [17]). Theorem 2 shows that \({\overline{C}} _{mNJ}^{(2)}(X)={\overline{C}}_{mNJ}^{(2)}(X^{*})\) whenever \(X=L^{p}(\mu )\) or \(X=l_{m}^{p}\) with \(m\ge 2^{n-1}.\) Really, fix \(1<p<2\) and consider \( X=L^{p}(\mu )\) or \(X=l_{m}^{p}\) with \(m\ge 2^{n-1}.\) Let q be conjugate to p. Then \(q>2\) and \(X^{*}=L^{q}(\mu )\) or \(X^{*}=l_{m}^{q}\) with \( m\ge 2^{n-1},\) respectively. Applying Theorem 2 for \( n=2,\) we have
Since \(q=\frac{p}{p-1}>2,\) it follows from Theorem 2 that
whence \({\overline{C}}_{mNJ}^{(2)}(X)={\overline{C}}_{mNJ}^{(2)}(X^{*}).\)
The equality \({\overline{C}}_{mNJ}^{(n)}(X)={\overline{C}} _{mNJ}^{(n)}(X^{*})\) does not hold in general for \(n\ge 3.\) By Remark \( 9 \,(ii)\) in [17] and Theorem 2, we have
whence \({\overline{C}}_{mNJ}^{(n)}(X)\not ={\overline{C}}_{mNJ}^{(n)}(X^{*})\) for \(n\ge 3.\)
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This research is supported by the grant 04/43/DSPB/0102 from Polish Ministry of Science and Higher Education.
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Communicated by Mikhail Ostrovskii.
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Ciesielski, M., Płuciennik, R. On some modifications of n-th von Neumann–Jordan constant for Banach spaces. Banach J. Math. Anal. 14, 650–673 (2020). https://doi.org/10.1007/s43037-019-00033-1
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DOI: https://doi.org/10.1007/s43037-019-00033-1
Keywords
- von Neumann–Jordan constant
- Modified n-th von Neumann–Jordan constant
- Uniformly non-\(l_{n}^{(1)}\)-Banach
- B-convexity