[PPT] - Convolution operators in discrete Ces` aro spaces Werner Ricker PowerPoint Presentation

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Convolution operators in discrete Ces` aro spaces

Werner Ricker Paweł Doma´ nski Memorial Conference Banach Center in Bedlewo Poland 1-7 July 2018

Werner Ricker Convolution in discrete Ces` aro spaces Doma´ nski Memorial Conference 0 / 12

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The Ces` aro operator C

Consider the Ces` aro operator C : CN → CN given by

C(x) ≔

x0, x0 + x1

2

, x0 + x1 + x2

3

, . . .

,

x = (xn)∞

0 ∈ CN.

|C(x)| ≤ C(|x|), ∀x ∈ CN (with |x| ≔ (|xn|)∞

0 )

C is a vector space isomorphism of CN onto CN.

The discrete Ces` aro space for 1 < p < ∞ (early 1970’s):

ces(p) ≔

        

x ∈ CN : xces(p) ≔



     

1 n+1 n

k=0

|xk|       

∞

p

= C(|x|)p < ∞         

Intense research by G. Bennett, Mem. Amer. Math. Soc.120 (1996): “Factorizing the classical inequalities”

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Properties of ces(p) , 1 < p < ∞

(ces(p), · ces(p)) is a reflexive Banach lattice.

ek ≔ (δnk)∞

n=0 , k ∈ N are an unconditional basis:

ekces(p) ≃ (k + 1)−1/p′,

k ∈ N,

1

p + 1 p′ = 1

.

Equivalent norms in ces(p): (a) x →

∞

k=n |xk| k+1

∞

n=0

p.

(b) x →

|x0|p + ∞

j=0 2j(1−p) 2j+1−1 k=2j |xk|

p1/p

. Hardy’s inequality for 1 < p < ∞ : C(|x|)p ≤ p′xp, x ∈ ℓp.

⇒ C maps ℓp → ℓp continuously (operator norm is p′).

Moreover, the inclusion ℓp ⊆ ces(p) is proper, continuous and C maps ces(p) → ℓp continuously (isometrically). Hence, also C maps ces(p) → ces(p) continuously.

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Further properties of ces(p), 1 < p < ∞

Let 1 < p < ∞ and x ∈ CN. Remarkable property (G. Bennett): x ∈ ces(p) if and only if C(|x|) ∈ ces(p)

ℓp, 1 < p < ∞, do not have this property.

G. Curbera (2014): The largest of all those solid Banach lat-

tices X ⊆ CN with ℓp ⊆ X s.t. C maps X → ℓp cont. is ces(p).

G. Curbera (2014): Largest amongst the class of spaces

ℓr (1 < r < ∞) satisfying ℓr ⊆ ces(p) is the space ℓp.

Dual Banach space ces(p)∗ identified by A.A. Jagers (1974). Rather complicated: G. Bennett (1996) showed: ces(p)∗ ≃ d(p′) =

x ∈ CN : ∞

n=0 supk≥n |xk|p′ < ∞

for the equivalent (but not dual) norm

xd(p′) ≔

(supk≥n |xk|)∞

n=0

p′ ≔
ˆ

x

p′ .

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Convolution operators in ces(p)

Fix b = (bn)∞

n=0 ∈ CN. Each element

x ∗ b ≔

       

n

j=0

xjbn−j

       

∞ n=0

,

x ∈ CN, again belongs to CN. So, we have the convolution operator Tb : x → x ∗ b = b ∗ x, which is well defined and linear from CN → CN. Moreover, TbTc = TcTb = Tb∗c, b, c ∈ CN. Relevant are the following identities: Te0 = I (Identity operator), i.e. x ∗ e0 = x, ∀x ∈ CN. en = e1 ∗ e1 ∗ . . . ∗ e1

(n terms), ∀n ≥ 1.

We first recall when Tb acts in the spaces ℓp, 1 < p < ∞.

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Convolution p-multipliers for ℓp, 1 ≤ p ≤ ∞

b ∈ CN is a (convolution) p-multiplier for ℓp (write b ∈ M (ℓp)) if x ∗ b ∈ ℓp,

∀x ∈ ℓp.

Closed graph theorem ⇒ Tb : ℓp → ℓp is continuous. Facts: [N.K. Nikolskii (1966) & (p = 2) I. Schur (1917)]

M (ℓ1) = M (ℓ∞) = ℓ1. M (ℓp) = M (ℓp′)

If 1 ≤ p1 < p2 ≤ 2, then M (ℓp1) M (ℓp2).

ℓ1 M (ℓp)

H∞, whenever 1 < p < ∞ (and M (ℓp) ℓp). Schur: M (ℓ2) = H∞, i.e., each element of M (ℓ2) is the sequence of Taylor coefficients of some function from H∞(D). What if we replace ℓp, 1 < p < ∞, with ces(p), 1 < p < ∞?

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Convolution p-multipliers for ces(p), 1 < p < ∞

Which elements b ∈ CN satisfy (for 1 < p < ∞ fixed) x ∗ b ∈ ces(p),

∀x ∈ ces(p)?

Equivalently, when does Tb map ces(p) → ces(p) continuously? Answer is dramatically different than for ℓp spaces. Proposition 1 [Curbera (2014)] Let 1 < p < ∞ and b ∈ CN. Then Tb : CN → CN maps ces(p) into ces(p), i.e. b ∈ M (ces(p)), if and only if b ∈ ℓ1. In this case

Tbop = bℓ1 =

∞

n=0

|bn| . Lp ≔ (L(ces(p)), ·op) is a unital, non-commutative Banach

algebra (for composition of operators from ces(p) → ces(p)). How does one identify the spectrum σ(Tb) of Tb ∈ Lp?

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Proposition 1 implies Mp ≔

Tb : b ∈ M (ces(p)) = ℓ1

is a unital, commutative, closed (proper) subalgebra of (Lp, ·op).

Mp is isometrically isomorphic to the B-algebra A ≔ (ℓ1, ∗).

Here, ∗ is convolution: the unit is e0 = (1, 0, 0, . . .).

R ⊆ C(D) is the unital, commutative B-algebra of functions ϕb(z) ≔

∞

n=0

bnzn, z ∈ D, for all b ∈ ℓ1. We use pointwise operations of scalar multiplication, addition and product: the norm is

ϕbR ≔ bℓ1 .

Unit of R is the constant function 1. As

ϕbϕc = ϕb∗c, the map

b → ϕb is an isometric B-algebra isomorphism of A onto R. For a B-algebra B with unit e and u ∈ B, define

σB(u) ≔ {λ ∈ C : u − λe not invertible} and ρB(u) ≔ C \ σB(u).

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The B-algebras Mp ≃ A ≃ R (1 < p < ∞) are all isometrically

isomorphic. So the spectrum of Tb satisfies

σMp(Tb) = σA (b) = σR(ϕb), ∀b ∈ ℓ1.

(⋆) Maximal ideal space of R is D. Gelfand theory and (⋆) imply:

σMp(Tb) = ϕb(D) = ϕb(z) : |z| ≤ 1 ,

b ∈ ℓ1 (1 < p < ∞).

Mp is a closed, unital subalgebra of the non-commutative unital

B-algebra Lp. Consequently,

σLp(Tb) ⊆ σMp(Tb),

b ∈ ℓ1. If the more traditional notation σ(Tb) is used for

σLp(Tb) =

λ ∈ C : (Tb − λI) not invertible in Lp
,

the previous containment becomes

σ(Tb) ⊆ ϕb(z) : |z| ≤ 1 ,

b ∈ ℓ1. How does one deduce this is an equality?

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The shift-operator in ces(p), 1 < p < ∞

The right-shift operator Sp in ces(p), given by Sp ((xn)∞

n=0) ≔ (0, x0, x1, . . .),

x ∈ ces(p), satisfies

Sp
p = 1 and the identity (with e1 = (0, 1, 0, 0, . . .))

Sp (x) = Te1(x) = x ∗ e1, x ∈ ces(p) . This formula, in turn, implies that Sn

p = Ten = Te1∗...e1,

(n times convolution.)

Via the isomorphism ces(p)∗ ≃ d(p′), the adjoint operator S∗

p : ces(p)∗ → ces(p)∗ can be identified with the left-shift operator

Lp′ in d(p′) given by u → Lp′ ((un)∞

n=0) ≔ (u1, u2, u3, . . .),

u ∈ d(p′).

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Spectrum of the operator Tb in ces(p)

Direct calculation: Every λ ∈ D is eigenvalue of S∗

p

with eigenvector xλ = (1, λ, λ2, λ3, . . .) ∈ d(p′).

⇒ D ⊆ σpt(S∗

p) = σ(Sp).

⇒ For all b = ∞

n=0 bnen ∈ ℓ1 we have

Tb = ∞

n=0 bnTen = ∞ n=0 bnSn p

(absolute conv. in ·op).

⇒ T ∗

b = ∞ n=0 bn(S∗ p)n

(absolute conv. in ·op). Since (S∗

p)nxλ = λnxλ, ∀n ∈ N, implies T ∗ bxλ = ϕb(λ)xλ, we have

ϕb(λ) ∈ σ(T ∗

b) = σ(Tb),

∀λ ∈ D.

This implies the reverse inclusion to yield:

Proposition 2.

Let 1 < p < ∞. Then, for each b ∈ ℓ1, we have

σ(Tb) = ϕb(D) =

ϕb(z) : z ∈ D
.

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Wiener’s theorem for the B-algebra R

If ϕb(z) 0 for all z ∈ D, then z → 1/ϕb(z), z ∈ D, belongs to R.

Then (1/ϕb) = ϕc for some c ∈ ℓ1. This yields Fact 1: Mp is an inverse-closed subalgebra of Lp. That is, if Tb − λI (with b ∈ ℓ1) is invertible in Lp, then

[(Tb − λI)−1 : ces(p) → ces(p)] = Tc

for some c ∈ ℓ1. Fact 2: The formula σ(Tb) = ϕb(D) = {ϕb(z) : z ∈ D} implies that Tb ∈ L fails to be compact ∀b ∈ ℓ1 \ {0}, 1 < p < ∞.

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Thank you for your attention!

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