Polynomial Numerical Index of l

° Kyungpook Mathematical Journalc

Polynomial Numerical Index of l

(1 < p < ∞)

Sung Guen Kim

Department of Mathematics, Kyungpook National University, Daegu 702-701, Korea e-mail : sgk317@knu.ac.kr

Abstract. We present some estimates for the polynomial numerical index of lp(1 < p <

∞).

1. Introduction

Given a Banach space E we write BEfor its unit ball and SEfor its unit sphere.

The dual space of E is denoted by E^∗ and let

Π(E) = {(x, x^∗) : x ∈ SE, x^∗∈ SE^∗, x^∗(x) = 1}.

A mapping P : E → E is called a (continuous) k-homogeneous polynomial if there is a (continuous) k-linear mapping A : E × · · · × E → E such that P (x) = A(x, · · · , x) for every x ∈ E. Let P(^kE : E) denote the Banach space of all k-homogeneous polynomials from E to itself, endowed with the polynomial norm kP k = sup_x∈BEkP (x)k. We refer to a book [6] by Dineen for background on polynomials. It is natural to generalize the concepts of numerical range and numerical radius of linear operators to homogeneous polynomials. The numerical range of P ∈ P(^kE : E) is defined to be the set of scalars

V (P ) := {x^∗(P x) : (x, x^∗) ∈ Π(E)}

and the numerical radius of P is defined by

v(P ) := sup {|λ| : λ ∈ V (P )}.

Received May 26, 2014; accepted October 23, 2014.

2010 Mathematics Subject Classification: Primary 46A22, 46G20.

Key words and phrases: Homogeneous polynomial, numerical radius, polynomial numerical index.

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2013R1A1A2057788).

615

Clearly, v(·) is a semi-norm on P(^kE : E), and v(P ) ≤ kP k for every P ∈ P(^kE : E).

As in the linear case, the author et al. [3] introduced the concept of the polynomial numerical index of order k of E to be the constant

n^(k)(E) := inf{v(P ) : P ∈ P(^kE : E), kP k = 1}

= sup{M ≥ 0 : kP k ≤ 1

Mv(P ) for all P ∈ P(^kE : E)}.

Of course, n⁽¹⁾(E) coincides with the usual numerical index of the space E. Note that 0 ≤ n^(k)(E) ≤ 1, and n^(k)(E) > 0 if and only if v(·) is a norm on P(^kE : E) equivalent to the usual norm. It is obvious that if E1, E2 are isometrically isomorphic Banach spaces, then n^(k)(E1) = n^(k)(E2).

The concept of the numerical index was first suggested by G. Lumer in 1968 (see [14]). The author et al. [3] introduced and studied the concept of the polyno- mial numerical index of order k of a Banach space, generalizing to k-homogeneous polynomials the classical numerical index. Recently, the author et al. [11] com- puted that ¹₂ = n⁽²⁾(c0) = n⁽²⁾(l1) = n⁽²⁾(l∞) for the real spaces c0, l1and l∞. Very recently, Garcia et al. [9] gave estimates for the polynomial numerical indices of C(K) and L1(µ). For general information and background on polynomial numerical index, we refer to ([1]–[4], [7]–[15]).

In this paper, we present some inequality for the norm of 2-homogeneous poly- nomials from the real space lp (1 < p < ∞) to itself in terms of their coefficients.

Using this we give a lower bound for n⁽²⁾(lp). We also present some upper bounds for n^(k)(lp).

2. Norm and Numerical Radius of 2-Homogeneous Polynomial on lp

Theorem 2.1. Let 1 < p < ∞, 1 = ¹_p + ¹_q and P (x, y) = (P1(x, y), P2(x, y)) ∈ P(²l²_p: l²_p), where Pk(x, y) = akx²+ bky²+ ckxy for k = 1, 2 and (x, y) ∈ l²_p. Then

(1) kP k^p≥ sup

0≤λ≤1|P1(λ^1p, (1 − λ)^1p)|^p+ |P2(λ^1p, (1 − λ)^1p)|^p; (2) v(P ) ≥ sup

0≤λ≤1|λ^1qP1(λ^1p, (1 − λ)^1p) + (1 − λ)^1qP2(λ^1p, (1 − λ)^1p)|.

Corollary 2.2. Let 1 < p < ∞, 1 = _p¹ +¹_q and P (x, y) = (P1(x, y), P2(x, y)) ∈ P(²l²_p: l²_p), where Pk(x, y) = akx²+ bky²+ ckxy for k = 1, 2 and (x, y) ∈ l²_p. Then

(1) kP k^p≥ sup

0≤λ≤1,k=1,2

{|P1([λ + (1 − λ)|ak− bk| (|ck|^p+ |ak− bk|^p)^1p]^1p,

[1 − λ − (1 − λ)|ak− bk|

(|ck|^p+ |ak− bk|^p)^1p]^1p)|^p+ |P2([λ + (1 − λ)|ak− bk| (|ck|^p+ |ak− bk|^p)^1p]^p1,

[1 − λ − (1 − λ)|ak− bk|

(|ck|^p+ |ak− bk|^p)^1p]^1p)|^p};

(2) v(P ) ≥ sup

0≤λ≤1,k=1,2{|[λ + (1 − λ)|ak− bk| (|ck|^p+ |ak− bk|^p)^1p]^1q P1([λ + (1 − λ)|ak− bk|

(|ck|^p+ |ak− bk|^p)^1p]^1p, [1 − λ − (1 − λ)|ak− bk| (|ck|^p+ |ak− bk|^p)^1p]^1p) + [1 − λ − (1 − λ)|ak− bk|

(|ck|^p+ |ak− bk|^p)^1p]^1q × P2([λ + (1 − λ)|ak− bk| (|ck|^p+ |ak− bk|^p)^1p]^p1, [1 − λ − (1 − λ)|ak− bk|

(|ck|^p+ |ak− bk|^p)^1p]^p1)|}.

Corollary 2.3. Let 1 < p < ∞, 1 = ¹_p +¹_q and P (x, y) = (P1(x, y), P2(x, y)) ∈ P(²l²_p: l²_p), where Pk(x, y) = akx²+ bky²+ ckxy for k = 1, 2 and (x, y) ∈ l_p². Then (1) kP k^p≥ sup

k=1,2

{|P1([1

p+ |ak− bk|

q(|ck|^p+ |ak− bk|^p)^1p]^1p, [1

q− |ak− bk|

q(|ck|^p+ |ak− bk|^p)^1p]^p1)|^p +|P2([1

p+ |ak− bk|

q(|ck|^p+ |ak− bk|^p)^1p]^p1, [1

q − |ak− bk|

q(|ck|^p+ |ak− bk|^p)^1p]^1p)|^p,

|P1([1

q − |ak− bk|

q(|ck|^p+ |ak− bk|^p)^1p]^1p, [1

p+ |ak− bk|

q(|ck|^p+ |ak− bk|^p)^1p]^1p)|^p +|P2([1

q − |ak− bk|

q(|ck|^p+ |ak− bk|^p)^1p]^1p, [1

p+ |ak− bk|

q(|ck|^p+ |ak− bk|^p)^1p]^1p)|^p};

(2) v(P ) ≥ sup

k=1,2{|[1

p+ |ak− bk|

q(|ck|^p+ |ak− bk|^p)^1p]^1qP1([1

p+ |ak− bk|

q(|ck|^p+ |ak− bk|^p)^p1]^1p, [1

q − |ak− bk|

q(|ck|^p+ |ak− bk|^p)^1p]^1p) + [1

q − |ak− bk|

q(|ck|^p+ |ak− bk|^p)^1p]^q1

×P2([1

p+ |ak− bk|

q(|ck|^p+ |ak− bk|^p)^1p]^1p, [1

q − |ak− bk|

q(|ck|^p+ |ak− bk|^p)^1p]^1p)|,

|[1

q− |ak− bk|

q(|ck|^p+ |ak− bk|^p)^1p]^1qP1([1

q− |ak− bk|

q(|ck|^p+ |ak− bk|^p)^1p]^p1, [1

p+ |ak− bk|

q(|ck|^p+ |ak− bk|^p)^1p]^1p) + [1

p+ |ak− bk|

q(|ck|^p+ |ak− bk|^p)^1p]^1q

×P2([1

q − |ak− bk|

q(|ck|^p+ |ak− bk|^p)^1p]^1p, [1

p+ |ak− bk|

q(|ck|^p+ |ak− bk|^p)^1p]^1p)|}

3. Estimates for the Polynomial Numerical Index of l_p

Theorem 3.1. Let 1 < p < ∞, 1 = ¹_p +¹_q and P (x) = (Pk(x))^∞_k=1 ∈ P(²lp : lp), where

Pk(x) =X

i≤j

a^(k)_ij xixj (x = (xn)^∞_n=1∈ lp).

Then

kP k ≤ k ( X∞ k=1

(|a^(k)₁₁| +1 2

X

1<j

|a^(k)_1j |), X∞ k=1

(|a^(k)₂₂| +1

2|a^(k)₁₂| +1 2

X

2<j

|a^(k)_2j |), · · · , X∞

k=1

(|a^(k)_nn| + 1 2

X

m<n

|a^(k)_mn| +1 2

X

n<j

|a^(k)_nj|), · · · ) kq.

Proof. It follows that

kP k = sup

x=(xn)∈S_lp

( X∞ k=1

|Pk(x)|^p)^1/p

≤ sup

x=(xn)∈S_lp

( X∞ k=1

( X∞ i=1

|a^(k)_ii ||xi|²+X

i<j

|a^(k)_ij ||xi||xj| )^p)^1/p

≤ sup

x=(xn)∈S_lp

X∞ k=1

( X∞ i=1

|a^(k)_ii ||xi|²+X

i<j

|a^(k)_ij ||xi||xj| )

≤ sup

x=(xn)∈S_lp

X∞ k=1

( X∞ i=1

|a^(k)_ii ||xi|²+X

i<j

|a^(k)_ij |(|x_i|²+ |x_j|²

2 ))

= sup

x=(x_n)∈S_lp

[ X∞ k=1

(|a^(k)₁₁| +1 2

X

1<j

|a^(k)_1j |)|x1|²

+ X∞ k=1

(|a^(k)₂₂| +1

2|a^(k)₁₂| +1 2

X

2<j

|a^(k)_2j |)|x2|²

+ · · · + X∞ k=1

(|a^(k)_nn| + 1 2

X

m<n

|a^(k)_mn| +1 2

X

n<j

|a^(k)_nj|)|xn|²+ · · · ]

≤ sup

x=(xn)∈S_lp

[ X∞ k=1

(|a^(k)₁₁| +1 2

X

1<j

|a^(k)_1j |)|x1|

+ X∞ k=1

(|a^(k)₂₂| +1

2|a^(k)₁₂| +1 2

X

2<j

|a^(k)_2j |)|x2|

+ · · · + X∞ k=1

(|a^(k)_nn| + 1 2

X

m<n

|a^(k)_mn| +1 2

X

n<j

|a^(k)_nj|)|x_n| + · · · ]

= k ( X∞ k=1

(|a^(k)₁₁| +1 2

X

1<j

|a^(k)_1j |), X∞ k=1

(|a^(k)₂₂| +1

2|a^(k)₁₂| +1 2

X

2<j

|a^(k)_2j |), · · · , X∞

k=1

(|a^(k)_nn| +1 2

X

m<n

|a^(k)_mn| +1 2

X

n<j

|a^(k)_nj|), · · · ) kq. 2

Theorem 3.2. Let 1 < p < ∞, 1 = ¹_p+¹_q. Then

n⁽²⁾(l_p) ≥

inf{ sup_n∈N(_n¹)^{1+ 1p}|P_n

l=1 P

i≤ja^(l)_ij| k (P_n

k=1(|a^(k)₁₁| +¹₂P_n

j=2|a^(k)_1j|), · · · ,P_n

k=1(|a^(k)_nn| +¹₂P

m<n|a^(k)_mn|), 0, 0, · · · ) kq :

nonzero P (x) = (P_k(x))^∞_k=1∈ P(²l_p: l_p), P_k(x) =X i≤j

a^(k)_ij x_ix_j}.

Proof. Suppose that P (x) = (Pk(x))^∞_k=1∈ P(²lp: lp) with Pk(x) =X

i≤j

a^(k)_ij xixj (x = (xn)^∞_n=1∈ lp).

Let n ∈ N, x0=P_n

k=1(_n¹)^1/pek, and x^∗₀=P_n

k=1(_n¹)^1/qek. Then (∗) v(P ) ≥ |x^∗₀(P (x0))| = (1

n)^1+p1| Xn l=1

X

i≤j

a^(l)_ij|.

It follows that by (∗) and Theorem 3.1, n⁽²⁾(lp)

= inf{v(P )

kP k : P (x) = (Pk(x))^∞_k=1∈ P(²lp: lp), P 6= 0}

≥ inf{ sup_n∈N(_n¹)^{1+ 1p}|P_n

l=1 P

i≤ja^(l)_ij| k ( P_n

k=1(|a^(k)₁₁| +¹₂P_n

j=2|a^(k)_1j|), · · · ,P_n

k=1(|a^(k)_nn| +¹₂P

m<n|a^(k)_mn|), 0, 0, · · · ) k_q :

nonzero P (x) = (Pk(x))^∞_k=1∈ P(²lp: lp)}. 2 Corollary 3.3. We have

inf{ sup_n∈N(_n¹)³2|P_n

l=1 P

i≤ja^(l)_ij| k (P_n

k=1(|a^(k)₁₁| +¹₂P_n

j=2|a^(k)_1j|), · · · ,P_n

k=1(|a^(k)_nn| +¹₂P

m<n|a^(k)_mn|), 0, 0, · · · ) k₂ :

nonzero P (x) = (Pk(x))^∞_k=1∈ P(²l2: l2), Pk(x) =P

i≤ja^(k)_ij xixj} = 0.

Theorem 3.4. Let k ∈ N with k ≥ 2. Then n⁽²⁾(l_k²) = inf{kQPk_l²

k: P ∈ P(²l²_k : l_k²),

kP k = 1, P (x, y) = (ax²+ by²+ cxy, a⁰x²+ b⁰y²+ c⁰xy)}, where QP ∈ P(^k+1l²_k) with QP(x, y) = ax^k+1+ bx^k−1y²+ cx^ky + a⁰x²y^k−1 +b⁰y^k+1+ c⁰xy^k.

Theorem 3.5. Let 1 < p < ∞ and k ∈ N with k ≥ 2. Then n^(k)(lp) ≤ n^(k)(l²_p)

≤ min{( p − 1

k + p − 1)^1−1p ( k

k + p − 1)^kp, (p + k − 1

2p + k − 2)^p1(_2p+k−2^p+k−2)^p+k−2p (_p+k−1^k−1 )^k−1p }.

Proof. Let P1(x, y) = (x^k, 0) and P2(x, y) = (x^k−1y, 0) for (x, y) ∈ l_p².

Then Pj∈ P(^kl²_p: l²_p) for j = 1, 2 and kP1k = 1. Some computation shows that kP2k = ( k − 1

p + k − 1)^k−1p ( p p + k − 1)^1p, v(P₂) = (p + k − 2

2p + k − 2)^p+k−2p ( p 2p + k − 2)^p1, and v(P1) = ( p − 1

k + p − 1)^1−1p ( k k + p − 1)^kp. It follows that

n^(k)(lp) ≤ n^(k)(l²_p) ≤ min{v(P1),v(P2) kP₂k}

= min{( p − 1

k + p − 1)^1−1p ( k

k + p − 1)^kp, ( p + k − 1

2p + k − 2)^1p(_2p+k−2^p+k−2)^p+k−2p (_p+k−1^k−1 )^k−1p },

which completes the proof. 2

Theorem 3.6. Let k ∈ N. Then n^(2k+1)(l2k) = 0 for the real space l2k. Proof. Let

P0(x) := (−x1x^2k₂ , x^2k₁ x2, 0, 0, . . .) (x = (xj) ∈ l2k).

Then kP0k 6= 0. Note that if (x1, x2, 0, 0, . . .) ∈ Sl_2k, then (x^2k−1₁ , x^2k−1₂ , 0, 0, . . .) ∈ S_(l2k)^∗= Sl 2k

2k−1

. It follows that v(P0)

= sup

x^2k₁ +x^2k₂ =1

| < (sign(x^2k₁ )x^2k−1₁ , sign(x^2k₂ )x^2k−1₂ , 0, . . .), P₀(x₁, x₂, 0, . . .) > |

= sup{| < (x^2k−1₁ , x^2k−1₂ , 0, 0, . . .), P0(x1, x2, 0, 0, . . .) > | : x^2k₁ + x^2k₂ = 1}

= sup{| − x1x^2k₂ + x1x^2k₂ | : x^2k₁ + x^2k₂ = 1} = 0.

Thus

n^(2k+1)(l2k) ≤ v(P0) kP0k = 0.

2 Corollary 3.7. Let k, s ∈ N with s ≥ 2k + 1. Then n^(s)(l2k) = 0 for the real space l2k.

Notice that there exist 1 < p < q < ∞, N ≥ 2 such that n^{(N )}(lp) = n^{(N )}(lq).

In fact, take p = 2k, q = 2(k + 1), N = 2k + 3. By Corollary 3.7, n^{(N )}(lp) = 0 = n^{(N )}(lq).

We are in position to prove our main result.

Theorem 3.8. Let 1 < p < ∞. Then for the complex space lp, n^(k)(lp) ≤ 2^1−kp (∀k ∈ N).

Proof. Simple computation shows that sup

|x1|^p+|x2|^p=1

(|x1|^p+k−1+ |x2|^p+k−1) = 2^1−kp = sup

|x1|^p+|x2|^p=1

(|x1|^kp+ |x2|^kp)^1p.

For every b ∈ C, define

Qb(x) := (x^k₁+ bx^k₂, bx^k₁+ x^k₂, 0, 0, . . .) ∈ P(^klp) (x = (xj) ∈ lp).

Then kQbk ≥ |Qb(1, 0)| = (1 + |b|^p)^1p and v(Qb)

= sup

|x1|^p+|x2|^p=1

| < (sign(x^p₁)x^p−1₁ , sign(x^p₂)x^p−1₂ , 0, 0, . . .), Qb(x1, x2, 0, 0, . . .) > |

≤ sup

|x1|^p+|x2|^p=1

(|x1|^p+k−1+ |x2|^p+k−1) +|b| sup

|x1|^p+|x2|^p=1

(|x1|^p−1|x2|^k+ |x2|^p−1|x1|^k)

≤ 2^1−kp + |b| sup

|x1|^p+|x2|^p=1

(|x1|^(p−1)q+ |x2|^(p−1)q)^1q sup

|x1|^p+|x2|^p=1

(|x1|^kp+ |x2|^kp)^1p (by Holder’s inequality)

= 2^1−kp + |b| sup

|x1|^p+|x2|^p=1

(|x₁|^p+ |x₂|^p)^q1 sup

|x1|^p+|x2|^p=1

(|x₁|^kp+ |x₂|^kp)^1p

= 2^1−kp + |b|2^1−kp

= 2^1−kp (1 + |b|).

It follows that

n^(k)(lp) ≤ inf

b∈C

v(Qb) kQbk ≤ inf

b∈C

2^1−kp (1 + |b|)

(1 + |b|^p)^1p = 2^1−kp .

2 Corollary 3.9. Let 1 < p < ∞. Then

(1) n^(k)(lp) ≤ 2^−1p < 1 (∀k ≥ 2).

(2) n^(k)(lp) ≤ ¹₂ (∀k ≥ p + 1).

(3) _2+M^2k

k(2^k−2) ≤ n^(k)(l1) ≤ (¹₂)^k−1, where k ≥ 1 and Mk=P_k

j=1 j^k j!(k−j)!. (4) limk→∞n^(k)(lp) = 0.

Theorem 3.10. Let p be an even number and k an odd number with k ≤ p − 1.

Then

n^(k)(lp) ≤ ( k

p − 1)^p−1−kk − ( k

p − 1)^{p−1−kp−1} , when lp is the real space.

Proof. We define

Q0(x) := (−x^k₂, x^k₁, 0, 0, . . .) ∈ P(^klp) (x = (xj) ∈ lp).

Then kQ0k = 1 and v(Q0)

= sup

x^p₁+x^p₂=1

| < (x^p−1₁ , x^p−1₂ , 0, 0, . . .), Q0(x1, x2, 0, 0, . . .) > |

= sup

x^p₁+x^p₂=1

| − x^p−1₁ x^k₂+ x^k₁x^p−1₂ |

≤ sup

x^p₁+x^p₂=1

|x1x2|^k− |x1x2|^p−1

= ( k

p − 1)^p−1−kk − ( k

p − 1)^{p−1−kp−1} , when |x1x2| = ( k

p − 1)^p−1−k1 . Thus

n^(k)(lp) ≤ v(Q0) kQ0k = ( k

p − 1)^p−1−kk − ( k

p − 1)^{p−1−kp−1} .

2 Theorem 3.11. Let p and s be even numbers with s < p − 1. Then

n^(s)(lp) ≤ min{(s − 1

p − 1)^p−ss−1 − (s − 1

p − 1)^p−1p−s, 2^−ps}, when lp is the real space.

Proof. By Theorem 3.10,

n^(s)(l_p) ≤ n^(s−1)(l_p) ≤ (s − 1

p − 1)^s−1p−s− (s − 1 p − 1)^p−1p−s.

We define

Q0(x) := (−x^s₂, x^s₁, 0, 0, . . .) ∈ P(^slp) (x = (xj) ∈ lp).

Then kQ0k = 1 and v(Q0)

= sup

x^p₁+x^p₂=1

| < (x^p−1₁ , x^p−1₂ , 0, 0, . . .), Q₀(x₁, x₂, 0, 0, . . .) > |

= sup

x^p₁+x^p₂=1

| − x^p−1₁ x^s₂+ x^s₁x^p−1₂ |

= sup

x^p₁+x^p₂=1

|x1|^p−1|x2|^s+ |x1|^s|x2|^p−1

≤ sup

x^p₁+x^p₂=1

2|x1x2|^s

= 2^−sp. Thus

n^(k)(lp) ≤v(Q0)

kQ0k = 2^−sp. 2

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