A study of Rhaly operators onDirichlet series with real

frequencies

Ronald Koh Joon Wei

School of Physical and Mathematical Sciences

2019

A study of Rhaly operators onDirichlet series with real

frequencies

Ronald Koh Joon Wei

School of Physical and Mathematical Sciences

A thesis submitted to the Nanyang Technological University

in partial fulfilment of the requirement for the degree of

Master of Science

2019

Abstract

In this thesis, we recall several important properties of the general Dirichlet seriesand characterize the entire Dirichlet series in the first two chapters.

We also recall the Cesaro operator, an operator which was defined based on theCesaro sum, and is heavily related to the Cesaro limits. A more general versionof the Cesaro operator; Rhaly operators, will also be defined and the study of itsproperties on the Hilbert space of entire Dirichlet series is the bedrock of this thesis.

Criterions will be provided for properties of Rhaly operators on the Hilbert space ofentire Dirichlet series like boundedness and invariance. Compactness and Schattenp-class inclusions will also be studied on a smaller class of entire Dirichlet series; thesmall weighted Hilbert space of Dirichlet series. An additional section will also studyinjectivity and the closed range property of bounded Rhaly operators.

1

Acknowledgements

The author would like to thank and acknowledge a few people here:

His supervisor, Associate Professor Chua Chek Beng for his kind and regular atten-tion throughout the course of his master’s candidature.

His co-supervisor, Dr Le Hai Khoi for his patient guidance and important insightsto materials that lead to the results in this thesis.

The love of his life, Valerie Evangelin Laurent for her constant mental support andnever-ending encouragement throughout the process of writing of this thesis.

His parents, whom have been his pillar of support, be it emotional or financial,throughout his life.

His friends Do Duc Tai, Ng Hong Wai, Johan Chrisnata and Renon Lim for theirsupport in coursework throughout his 18 months as a master’s student. Credits toHong Wai for his inspiration in proving Theorem 3.25 in this thesis.

The members of the biweekly Analysis Seminar (chaired by Dr Le Hai Khoi), RenonLim, Camille Mau, Wee Jun Jie, etc for their attention during the author’s talk aboutthis thesis in early September, and subsequent comments that led to improvementsin this thesis.

2

Contents

1 Introduction 41.1 Classical Dirichlet series . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 General Dirichlet series . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Entire Dirichlet series 92.1 Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Characterization of entire Dirichlet series . . . . . . . . . . . . . . . . 11

3 Rhaly operators on the Hilbert space of entire Dirichlet series 123.1 Definition of the Rhaly operator Ra . . . . . . . . . . . . . . . . . . . 143.2 Preliminary definitions and results . . . . . . . . . . . . . . . . . . . . 153.3 Properties of the Rhaly operator Ra . . . . . . . . . . . . . . . . . . . 183.4 Closed range characterization of Ra . . . . . . . . . . . . . . . . . . . 22

4 Conclusion and Future Work 30

Bibliography 31

3

Chapter 1

Introduction

1.1 Classical Dirichlet series

We begin this thesis with a definition of the classical/ordinary Dirichlet series.

Definition 1.1. A classical/ordinary Dirichlet series is the series of the form

∞∑n=1

annz

(1.1)

where (an)∞n=1 is a complex sequence and z is a complex number.

The ordinary Dirichlet series plays a prominent role in analytic number theoryand has been a prime topic of study for many mathematicians ever since itsinception [1].

One of the main problems of the current century includes the solving of thenon-trivial zeroes of the Riemann zeta function, i.e. a specific case of the clas-sical Dirichlet series when an = 1 for all n,

ζ(z) :=∞∑n=1

1

nz,

the problem is also known as the Riemann Hypothesis : are the non-trivialzeroes of the Riemann zeta function on the line Re(z) = 1

2?

The convergence of the classical Dirichlet series is an interesting topic. Wefocus on the specific case where (an) and z are real instead. Suppose (1.1) isconvergent at some x0 ∈ R. Then rewriting (1.1) as

∞∑n=1

annz

=∞∑n=1

annx0· 1

nz−x0,

4

we observe that the sequence

bn :=1

nz−x0

is monotonically decreasing and hence (1.1) is convergent for z > x0 by Abel’stest of convergence. Note that z > x0 is the ”right hand side” of the realnumber line. The question then is, can we obtain something similar in a moregeneral case?

1.2 General Dirichlet series

We begin this section with the definition of the general Dirichlet series.

Definition 1.2. Let an and z = σ + it be complex numbers and (λn) be asequence of strictly increasing real positive numbers (the frequencies of theseries) where λn →∞ as n→∞ (denoted as 0 ≤ (λn) ↑ ∞). Then the series

∞∑n=1

ane−λnz (1.2)

is called a Dirichlet series with real frequencies.

A point to note is that (1.2) can be clearly seen as a more general case of (1.1),i.e. λn = log n.

From this point onward, the term “Dirichlet series” shall be referred to asthe general Dirichlet series. For Chapter 1 and 2, the proofs are deferred asthe focus is mainly on the study of Rhaly operators on the Hilbert space ofDirichlet series (Chapter 3). Most of the proofs can be found in [4, 12].

The following result describes the convergence of the Dirichlet series:

Proposition 1.3. Suppose the series

∞∑n=1

ane−λnz

converges at z = z0. Then it is uniformly convergent in the domain

{z ∈ C : | arg(z − z0)| ≤ θ <π

2}.

5

The preceding result tells us that the convergence of the Dirichlet series is ahalf-plane, and it can be reformulated as

Proposition 1.4. If the series

∞∑n=1

ane−λnz

converges at z0 = σ0 + it, then it is convergent for any z with σ > σ0.

Similar to the radius of convergence in Taylor’s series, we also have the abscissaof convergence of a Dirichlet series:

Definition 1.5. The abscissa of convergence, σc is defined as

σc = inf{σ ∈ R : (1.2) converges for any z such that Re(z) > σ}.

Definition 1.6. The half-plane of convergence is defined as

Cσc = {z ∈ C : Re(z) > σc}.

Remark 1.7. For the sake of consistency, we also consider C and ∅ as half-planes, i.e. C−∞ = C and C∞ = ∅.The following series are examples of Dirichlet series with their abscissas ofconvergence.

Example 1.8. By choosing λn = log n and an = 1n!

, the series

∞∑n=1

ane−λnz =

∞∑n=1

1

n!nz

is convergent for all values of z and hence σc = −∞.

Example 1.9. By choosing λn = log n and an = n!, the series

∞∑n=1

ane−λnz =

∞∑n=1

n!

nz

is convergent for no values of z and hence σc =∞.

Example 1.10. By choosing λn = log n and an = 1, the series

∞∑n=1

ane−λnz =

∞∑n=1

1

nz

is convergent for values of z with Re(z) > 1. Hence σc = 1.

6

Example 1.11. The series∞∑n=1

1

ne−nz

at z = −πi is the alternating harmonic series which is convergent, while onthe same line σ = 0, the series is divergent at z = 0. Hence, σc = 0.

Example 1.12. The series∞∑n=2

1

(log n)2· 1

nz

has an abscissa of convergence σc = 1, while it is convergent on all points onσ = 1.

In the following two results, we provide two different formulae to calculate theabscissa of convergence.

Proposition 1.13. If∑∞

k=1 ak diverges, then σc can be calculated via the fol-lowing formula

σc = lim supn→∞

log |An|λn

,

where

An =n∑k=1

ak.

Proposition 1.14. If∑∞

k=1 ak is convergent, then σc can be calculated via thefollowing formula

σc = lim supn→∞

log |Rn|λn+1

,

where

Rn =∞∑

k=n+1

ak.

We also have a notion of absolute convergence and abscissas of uniform andabsolute convergence of Dirichlet series.

Definition 1.15. A Dirichlet series if absolutely convergent if the series

∞∑n=1

|ane−λnz| =∞∑n=1

|an|e−λnσ (1.3)

is convergent.

7

Definition 1.16. The abscissa of absolute convergence, σa is defined as

σa = inf{σ ∈ R : (1.2) converges absolutely for any z such that Re(z) > σ}.

Definition 1.17. The abscissa of uniform convergence, σu is defined as

σu = inf{σ ∈ R : (1.2) converges uniformly for any z such that Re(z) > σ}.

It is clear that absolute convergence implies uniform convergence by the Weier-strass M-test, and also uniform convergence implies convergence. Hence σc ≤σu ≤ σa.

8

Chapter 2

Entire Dirichlet series

The aim of this chapter is to characterize some special Dirichlet series whichhas the abscissa of convergence σc = −∞, i.e. converges everywhere on thecomplex plane/is an entire function. We begin this chapter with a result thatgives an upper bound to the width of the strip of conditional convergence, thestrip on the complex plane where the Dirichlet series is conditionally conver-gent. As mentioned in Chapter 1, the most of the results and proofs in thischapter can be found in [4] and [12].

2.1 Preliminary results

Proposition 2.1. For the series (1.2), we have the inequality

0 ≤ σa − σc ≤ lim supn→∞

log n

λn=: L.

This quantity L is essential in characterizing the entire Dirichlet series, as wewill see later.

We also have a lower bound for the abscissa of convergence (which is an im-portant quantity to be seen in this chapter).

Proposition 2.2. For the series (1.2), we have a lower bound for the abscissaof convergence

σc ≥ lim supn→∞

log |an|λn

. (2.1)

The following result, also known as the Valiron formula [12], is the most im-portant result of this chapter.

9

Proposition 2.3 (Valiron forumla, [12]). Let L be defined as it was in Propo-sition 2.1. Suppose L is finite, then this chain of inequalities will always holdfor (1.2).

lim supn→∞

log |an|λn

≤ σc ≤ σu ≤ σa ≤ lim supn→∞

log |an|λn

+ L.

The Valiron formula is the most important result in helping us characterize apart of the entire Dirichlet series, i.e

Proposition 2.4. If L, as defined in Proposition 2.1, is finite, then the ab-scissa of convergence, abscissa of uniform convergence and the abscissa ofabsolute convergence of (1.2) are σc = σu = σa = −∞, i.e. (1.2) is convergenteverywhere on C if and only if

lim supn→∞

log |an|λn

= −∞. (2.2)

Example 2.5. Consider the series in Example 1.8, where an = 1n!

and λn = log n,

∞∑n=1

1

n!nz.

Note that

L = lim supn→∞

log n

λn= lim

n→∞

log n

log n= 1 <∞.

Now,

lim supn→∞

log |an|λn

= limn→∞

log 1n!

log n= −∞.

So by Proposition 2.4, the series in Example 1.8 represents an entire Dirichletseries.

We also note characterizing entire functions is possible with L < ∞; on theother hand, it is also possible to have entire functions when L =∞.Example 2.6. Consider the series

∞∑n=2

ane−λnz

where λn = log log n, an = 1n!

. We have

L = lim supn→∞

log n

λn= lim

n→∞

log n

log log n=∞.

10

Note that∞∑n=2

ane−λnz =

∞∑n=2

1

n!· e− log lognz =

∞∑n=2

1

n!· 1

(log n)z.

By applying the Ratio Test, we have

∞∑n=2

ane−λnz

converges absolutely for all z. That is to say that the series above is an entirefunction. So when L =∞, it is still possible to have entire functions.

2.2 Characterization of entire Dirichlet series

We continue this chapter by recalling a classical result from complex analysis.This result can be found in any textbook for elementary complex analysis.

Proposition 2.7. Suppose (fn) is a sequence of functions, all analytic on anopen set U . If (fn) converges uniformly to f on every compact subset of U ,then f is also analytic on U .

The result above is followed by a result of uniqueness of the Dirichlet series,i.e. if the two Dirichlet series is absolutely convergent for an abscissa largeenough and agree on a sequence tending to infinity, then the coefficients of thetwo series are the same.

Proposition 2.8 ([1]). Suppose

f(z) =∞∑n=1

ane−λnz, g(z) =

∞∑n=1

bne−λnz

are absolutely convergent for all σ > σa. Define (zk) = (σk + itk) to be asequence such that (σk) → ∞ as k → ∞ and let f(zk) = g(zk) for all k, thenan = bn for all n.

We end of this chapter with a corollary on the unique characterization of theentire Dirichlet series.

Corollary 2.9. The series (1.2), with the assumption that L < ∞, uniquelyrepresents an entire function if and only if

lim supn→∞

log |an|λn

= −∞.

11

Chapter 3

Rhaly operators on the Hilbertspace of entire Dirichlet series

Let 0 ≤ (λn) ↑ ∞ be a sequence of real numbers. Recall from Chapter 2(Corollary 2.9) that if this sequence satisfies the condition

lim supn→∞

log n

λn<∞,

then the Dirichlet series∑∞

n=1 cne−λnz converges in C (that is, it uniquely

defines an entire function) if and only if

lim supn→∞

log |cn|λn

= −∞.

We define a sequence space E as follows

E =

{a = (an) : lim sup

n→∞

log |an|λn

= −∞

}.

This sequence is associated with the inner product space

D ={f(z) =

∞∑n=1

cne−λnz : lim sup

n→∞

log |cn|λn

= −∞}

under the inner product

〈∞∑n=1

ane−λnz,

∞∑n=1

bne−λnz〉 =

∞∑n=1

anbn.

12

However, D is not a Hilbert space under the norm generated by this innerproduct. The inner product can be modified so that we obtain a subspace ofD which is a Hilbert space; for each sequence of real positive numbers β = (βn)(which we term as the weights), we associate it with the following sequencespace

Xβ =

{a = (an) : ‖a‖Xβ :=

(∞∑n=1

|an|2β2n

)1/2

<∞

},

which is a Hilbert space [5] with the inner product

〈a, b〉 =∞∑n=1

anbnβ2n. (3.1)

Based on the sequence of positive real numbers β, we define a function spaceof weighted entire Dirichlet series

H2(E, β) :=

{f(z) =

∞∑n=1

ane−λnz entire : ‖f‖H2(E,β) := ‖(an)‖Xβ <∞

}.

There is a relationship between Xβ and E, as described in the first elementaryresult below. First, we define the quantities

β∗ := lim infn→∞

log βnλn

and β∗ = lim supn→∞

log βnλn

.

Proposition 3.1 ([5]). There are only three possibilities relating Xβ and E:

1. Xβ ( E ⇐⇒ β∗ =∞,

2. E ( Xβ ⇐⇒ β∗ <∞,

3. E \ Xβ 6= ∅ and Xβ\ 6= ∅ ⇐⇒ β∗ < β∗ = ∞, which means that thespaces never coincide.

Note that the space H2(E, β) may not necessarily be a Hilbert space, but aresult in [5] provides a criterion on β for which H2(E, β) is a Hilbert space.

Proposition 3.2. Let 0 ≤ (λn) ↑ ∞ be a sequence of real numbers. Then thespace of entire Dirichlet series H2(E, β) is a Hilbert space if and only if thefollowing condition holds

β∗ = lim infn→∞

log βnλn

=∞. (3.2)

13

In this thesis, we adopt the convention that β∗ = ∞, so that H2(E, β) is aHilbert space. Under the convention that β∗ = ∞, Result 3.1 asserts thatXβ ( E, which means that H2(E, β) ( D. Now, we see that the originaldefinition of H2(E, β) can be reformulated as

H2(E, β) :=

{f(z) =

∞∑n=1

ane−λnz : (an) ∈ Xβ

}. (3.3)

3.1 Definition of the Rhaly operator Ra

Let c = (cn) be a complex sequence and let

sn(c) :=n∑k=1

ck

be the nth partial sum of the first n entries of c. The Cesaro operator C wasfirst introduced in [3] and it is defined to be

C(c) =(snn

)∞n=1

.

C is widely studied in various contexts in mathematics as it is related to thetheory of Cesaro limits (see, e.g. [3]).

Let a = (an) be a given complex sequence. Rhaly [8] generalized the Cesarooperator by defining a new operator

Ra =

a1 0 0 0 · · ·a1 a2 0 0 · · ·a1 a2 a3 0 · · ·a1 a2 a3 a4 · · ·...

......

.... . .

We note that the Cesaro operator is a specific case of Rhaly operators, namely

an =1

n, n ∈ N.

Rhaly operators acting on `p and other sequence spaces have been studied invarious papers (see e.g [7], [8] and the references therein). However, Rhaly

14

operators on function spaces have not been studied extensively; so far, only[11] addresses Rhaly operators on small weighted Hardy spaces.

We know from [11] that there are many properties of Rhaly operators onthe space of analytic functions on the unit disk D and small weighted Hardyspaces like invariance, boundedness, etc. The aim of this thesis is to study theproperties of Rhaly operators on H2(E, β).

Definition 3.3. For a Dirichlet series

f(z) =∞∑n=1

cne−λnz

and a complex sequence a = (an), the Rhaly operator induced by a is definedas

Ra(f)(z) =∞∑n=1

(an

n∑k=1

ck

)e−λnz. (3.4)

3.2 Preliminary definitions and results

In this section, we cover the required definitions and results required for thenext section. These definitions, results as well as the proofs of the results canbe found in the references [9, 10, 13].

Definition 3.4. Let X and Y be normed vector spaces. A linear operatorT : X → Y is said to be bounded if there exists an M > 0 such that for allx ∈ X, we have ‖T (x)‖ ≤ M ‖x‖ . The infimum of such M is the operatornorm of T, denoted as ‖T‖op , or simply ‖T‖ .

Definition 3.5. Let V,W be Hilbert spaces. For an operator T : V → W ,the adjoint of T , denoted as T ∗ is the operator T : W → V such that

〈T (v), w〉W = 〈v, T ∗(w)〉V

for all v ∈ V and w ∈ W.

For each w ∈ W , note that by defining the linear functional lw(v) = 〈T (v), w〉W ,we can obtain a unique u ∈ V such that

lw(v) = 〈T (v), w〉W = 〈v, u〉V .

By defining T ∗(w) as u, the adjoint operator T ∗ can be defined by using theRiesz Representation Theorem for each w ∈ W.

15

Definition 3.6. Let H be a Hilbert space. A bounded linear operator T :H → H is called a Hilbert-Schmidt operator if for any orthonormal basis{ej}j∈J of H,

‖T‖HS :=

(∑j

‖T (ej)‖2)1/2

<∞.

‖T‖HS is known as the Hilbert-Schmidt norm.

Example 3.7. Let (ej)∞j=1 be the canonical basis for the Hilbert space of square

summable sequences (see Example 3.7), i.e.

ej = (0, ..., 0, 1, 0, ...),

where 1 is in the jth coordinate of the sequence.Define an operator T : l2(R)→ l2(R) such that

T (x) =∞∑n=1

1

n〈x, en〉en,

where 〈, 〉 denotes the standard dot product. T is clearly linear and for anyi ∈ N,

T (ei) =1

iei

and hence

‖T‖HS =∞∑i=1

‖T (ei)‖2 =∞∑i=1

1

i2<∞.

Hence, T is Hilbert-Schmidt.

Example 3.8. On the other hand, consider another operator T1 on l2(R) suchthat for all x ∈ l2(R),

T1(x) =∞∑n=1

1√n〈x, en〉en

where (ej)∞j=1 is the standard basis of l2(R) and 〈, 〉 denotes the standard dot

product as in Example 3.7. Using techniques similar to Example 3.7, we cansee that T1 is not Hilbert-Schmidt as

∞∑n=1

1

n

is the divergent harmonic series.

16

Definition 3.9. Let H be a Banach space. A bounded linear operator T :H → H is said to be compact if T maps the unit ball of H to a relativelycompact subset of H, i.e. a subset of H which has compact closure.

Example 3.10. A Hilbert-Schmidt operator is compact (see the next proposi-tion).

Proposition 3.11. Let H be a Hilbert space and T : H → H be a Hilbert-Schmidt operator. Then T is compact.

It is also well-known that a Hilbert space is separable if and only if it hasa countable orthonormal basis. Henceforth, we define the Schatten p-classoperators on separable Hilbert spaces.

Definition 3.12. Suppose H is a separable Hilbert space. For 1 ≤ p < ∞,the Schatten p-class consists of all bounded linear operators T : H → H suchthat the Schatten p-norm ‖T‖p is finite, i.e.

‖T‖p :=

(∞∑n=1

spn(T )

) 1p

<∞.

Here, sn(T ) refers to the nth eigenvalue of the Hermitian operator |T | :=√T ∗T .

Example 3.13. The Hilbert-Schmidt class of operators is a particular case ofthe Schatten p-class operators, precisely when p = 2:

Let {si(T )}∞j=1 be the eigenvalues of |T |, then {s2i (T )}∞j=1 are eigenvalues forT ∗T. Let {ei}∞j=1 be the set of corresponding orthonormal eigenvectors of T ∗Twhich forms an orthonormal basis of H. Then

‖T‖HS =

(∞∑j=1

‖T (ej)‖2)1/2

=

(∞∑j=1

〈T (ej), T (ej)〉

)1/2

=

(∞∑j=1

〈T ∗T (ej), ej〉

)1/2

=

(∞∑j=1

s2j(T )

) 12

= ‖T‖2 .

There is an important property of the Schatten p-class of operators, the de-creasing nature of its norm as exemplified by the result below which we willnot prove as it is a standard textbook exercise.

17

Proposition 3.14. For any sequence of real numbers x = (xn)∞n=1 and p, q arereal numbers such that 1 ≤ q < p <∞,

‖x‖p :=

(∞∑n=1

|xn|p) 1

p

≤

(∞∑n=1

|xn|q) 1

q

=: ‖x‖q .

We now have all we need for the next section, where we study propertiesof the Rhaly operators Ra on the Hilbert space of entire Dirichlet series likeinvariance, boundedness, compactness and closed range.

3.3 Properties of the Rhaly operator Ra

The first question we should ask is: under what conditions does Ra(f)(z)converge on the whole complex plane C (i.e. represents an entire function)?

Theorem 3.15. Ra(f) is invariant on D if and only if a ∈ E.

Proof. First, suppose that Ra(f) is invariant on D. Let

f(z) =∞∑n=1

cne−λnz

such that c1 = 1 and ck = 0 for all k ≥ 2, i.e. f(z) = c1e−λ1z. Clearly f is in

D. We now note that for every n ≥ 1,

an

n∑k=1

ck = an,

hence

Ra(f)(z) =∞∑n=1

(an

n∑k=1

ck

)e−λnz =

∞∑n=1

ane−λnz

is in D by invariance assumption. Hence a ∈ E.Conversely, suppose that a ∈ E. Let

f(z) =∞∑n=1

cne−λnz ∈ D.

Define bn = an∑n

k=1 ck. It suffices to show that

lim supn→∞

log |bn|λn

= −∞.

18

Since f ∈ D, this implies that f(0) =∑∞

n=1 cn is convergent.

Since∑∞

n=1 cn is convergent, let

γ :=∞∑n=1

cn.

Now, there exists an N ∈ N such that for all n ≥ N, we have

|n∑k=1

ck − γ| < 1.

Then for all n ≥ N,

log |bn|λn

=log |an|λn

+log |

∑nk=1 ck − γ + γ|λn

≤ log |an|λn

+log(1 + |γ|)

λn.

Clearly, the second term of the right hand side of the inequality goes to 0,hence we also have the required result by taking the limit superior on bothsides of the inequality. Therefore,

lim supn→∞

log |bn|λn

= −∞,

and hence Ra(f) ∈ D, proving invariance of the Rhaly operator Ra.

As mentioned earlier, we will focus our study of Rhaly operators on the Hilbertspace H2(E, β). First, the following theorem encapsulates invariance of Ra onH2(E, β).

Theorem 3.16. The Rhaly operator Ra is invariant on H2(E, β) if and onlyif a ∈ Xβ.

Proof. Suppose Ra is invariant on H2(E, β). Then Ra(f)(z) ∈ H2(E, β) forany f ∈ H2(E, β).

Define

f(z) =∞∑k=1

cke−λkz

where c1 = 1 and ck = 0 for all k ≥ 2. Clearly f is in H2(E, β), since

‖(cn)‖2 =∑|cn|2β2

n = β21 <∞.

19

Then for this particular f ,

Ra(f)(z) =∞∑n=1

ane−λnz ∈ H2(E, β),

so a ∈ Xβ.

Conversely, suppose a ∈ Xβ. Let

f(z) =∞∑n=1

cne−λnz ∈ H2(E, β).

Clearly f(0) =∑∞

n=1 cn is convergent, since f is in H2(E, β). Hence for anyn ∈ N, the sequence of partial sums

sn :=n∑k=1

ck

are bounded. Therefore, there exists an M > 0 such that for all n ∈ N, wehave

|sn|2 =∣∣∣ n∑k=1

ck

∣∣∣2 ≤M.

As a result,

‖Ra(f)‖2 =∞∑n=1

|an|2β2n

∣∣∣ n∑k=1

ck

∣∣∣2 ≤M ·∞∑n=1

|an|2β2n <∞.

Hence Ra(f) is invariant on H2(E, β).

Theorem 3.17. Ra is bounded on H2(E, β) if and only if a ∈ Xβ.

Proof. Suppose a ∈ Xβ. We can see from the proof of Theorem 3.16 that Ra

is bounded on H2(E, β).Now suppose that Ra is bounded on H2(E, β). Then there exists an M > 0such that ‖Ra(f)‖ ≤ M ‖f‖ for all f ∈ H2(E, β). This implies that Ra isinvariant, and hence by Theorem 3.16, a ∈ Xβ.

Now, let us suppose that the sequence β satisfies the condition

∞∑n=1

1

β2n

<∞. (3.5)

20

Under this condition, βn is assumed to be “large”. Since we also require

∞∑n=1

|an|2β2n <∞,

this loosely means that |an| has to be small in order to compensate for the sumbeing finite. To add on to the above, under this condition, we are only lookingat a relatively small subset of the Hilbert space of entire Dirichlet series, whichwe are going to term as the small weighted Hilbert space of entire Dirichletseries. From now on, we will refer the small weighted Hilbert space of entireDirichlet series as H2(E, β) satisfying condition (3.5).

Also, H2(E, β) has a countable orthonormal basis (i.e. { 1βke−λkz}∞k=1), it follows

that H2(E, β) is separable. Hence it makes sense to define the Schatten p-classinclusions for operators on H2(E, β).

Theorem 3.18. The following statements for the operator Ra on the smallweighted Hilbert space of entire Dirichlet series are equivalent:

1. Ra belongs to Schatten p-class for all p ≥ 2,

2. Ra is Hilbert-Schmidt,

3. Ra is compact,

4. a ∈ Xβ.

Proof. (1)⇐⇒ (2) The Schatten 2-class is the Hilbert-Schmidt class of opera-tors (see Example 3.13) and also the Schatten norms are decreasing in nature(see Proposition 3.14); i.e if T is a compact linear operator on a Hilbert spaceH and 1 ≤ p ≤ q ≤ ∞, then ‖T‖p ≥ ‖T‖q .

(2) =⇒ (3) Hilbert-Schmidt operators are compact operators. (Proposition3.11)

(3) =⇒ (4) Compact operators are assumed to be bounded. This result thenfollows from Theorem 3.17.

(4) =⇒ (2) Suppose a ∈ Xβ. Consider the orthonormal basis {ek}∞k=1 forH2(E, β) where ek = β−1k e−λkz. It is easy to see that for each k ≥ 1,

ek =∞∑j=1

cje−λjz

21

where ck = β−1k and cj = 0 whenever j 6= k. Therefore, for each k,

Ra(ek)(z) =∞∑n=k

an1

βke−λnz.

Now,

‖Ra(ek)‖2 =∞∑n=k

|an|21

β2k

β2n =

1

β2k

∞∑n=k

|an|2β2n.

which gives

‖Ra‖2HS =∞∑k=1

‖Ra(ek)‖2 =∞∑k=1

1

β2k︸ ︷︷ ︸

<∞, due to (3.5)

·∞∑n=k

|an|2β2n <∞.

Hence Ra is Hilbert-Schmidt.

We summarize the above results in the following theorem.

Theorem 3.19. Let Ra be the Rhaly operator induced by a complex sequencea = (an). Then the following assertions are equivalent:

1. a ∈ Xβ,

2. Ra is invariant on H2(E, β),

3. Ra is bounded on H2(E, β),

Moreover, for the small weighted Hilbert space of entire Dirichlet series, (1)-(3)are equivalent to the following statements

4. Ra is compact,

5. Ra is in the Schatten p-class for p ≥ 2.

3.4 Closed range characterization of Ra

In this section, we characterize the closed range property of bounded Rhalyoperators. Unless otherwise stated, the Rhaly operator’s domain in this sectionis assumed to be H2(E, β). We begin this section with two definitions regardingfinite and strictly nonzero sequences.

22

Definition 3.20. Let a = (an)∞n=1 be a complex sequence. We say that a isan finite sequence if there exists an N ∈ N such that for all n ≥ N, an = 0.

In this definition, we adopt the convention that the zero sequence is a finitesequence.

Definition 3.21. Let a = (an)∞n=1 be a complex sequence. We say that a is astrictly nonzero sequence if an 6= 0 for all n ∈ N.

Suppose we have a sequence a = (an)∞n=1 that is not strictly nonzero. Thenthere exists an n0 ∈ N such that an0 = 0. Looking at the subsequent indicesof the sequence after n0, we have that either all the subsequent terms afteran0 are zero, that is, for all n > n0, an = 0, i.e. a is a finite sequence, or thatthere exists a subsequent term after an0 that is nonzero; there exists an n1 ∈ Nsuch that n1 ≥ n0 and an1+1 6= 0, which in this case, we say that a has a finitegap of zeroes. We also note that it is also possible for sequences that are notstrictly nonzero to have the two properties mentioned above.

Using strict nonzero-ness of the sequence a, we can characterize the injectivityof Ra as we will see in the following theorems.

Theorem 3.22. Suppose Ra is bounded. If a is a strictly nonzero sequence,then Ra is injective.

Proof. Assume that a is a strictly nonzero sequence and let

f(z) =∞∑n=1

cne−λnz, g(z) =

∞∑n=1

dne−λnz

be in H2(E, β) such that Ra(f) = Ra(g). Since Ra is bounded, it follows fromTheorem 3.19 that Ra is invariant on H2(E, β). Therefore Ra(f) ∈ H2(E, β)and by uniqueness of entire Dirichlet series, we have

an

n∑k=1

ck = an

n∑k=1

dk

for all n ∈ N. Since an 6= 0 for all n ∈ N, this yields

n∑k=1

ck =n∑k=1

dk

for all n ∈ N. By induction, ck = dk for all k and hence f = g. Therefore Ra

is injective.

23

It also turns out that strict nonzero-ness of a is a necessary condition for Ra

to be injective, as exemplified by the following theorem.

Theorem 3.23. If Ra is injective, then a is a strictly nonzero sequence.

Proof. From the comments directly after Definition 3.21, we show that if a iseither a finite sequence or has a finite gap of zeroes, then Ra is not injective,i.e. we can find two Dirichlet series in H2(E, β)

f(z) =∞∑n=1

cne−λnz, g(z) =

∞∑n=1

dne−λnz

such that Ra(f) = Ra(g) but f 6= g.

Assume first that a is not a strictly nonzero sequence, that is, there exists ann0 ∈ N such that an0 = 0. Then at least one of these two cases will occur:

Case 1: For all n ≥ n0, an = 0.

In this case, consider two Dirichlet series

f(z) =∞∑n=1

cne−λnz, g(z) =

∞∑n=1

dne−λnz

such that ck = dk = 0 for k 6= n0, n0 + 1, and cn0 = dn0+1 = 0, cn0+1 = dn0 = 1.It is clear that f, g are both in H2(E, β) and f 6= g. However, for all n ∈ N,we have

an

n∑k=1

ck = 0 = an

n∑k=1

dk,

hence Ra(f) = Ra(g). This shows that Ra is not injective.

Case 2: ∃n1 ∈ N, n1 ≥ n0 such that ak = 0 for all n0 ≤ k ≤ n1 and an1+1 6= 0.

In this case, we also consider two Dirichlet series

f(z) =∞∑n=1

cne−λnz, g ≡ 0

where cn0 = 1, cn1+1 = −1 and ck = 0 otherwise. Then it is also clear thatf, g ∈ H2(E, β) and f 6= g. Similarly to Case 1, Ra(f) = Ra(g) = 0, and henceRa is not injective.

24

We note that in Theorem 3.23, boundedness of Ra is not needed as the rangeof Ra is not important; we only need to show that Ra maps two differentfunctions to the zero function, which is obviously in the range of Ra since Ra

is linear.

Summarizing Theorem 3.22 and 3.23, we can characterize injectivity for boundedRhaly operators in the following theorem.

Corollary 3.24. Suppose Ra is bounded. Then Ra is injective if and only ifa is a strictly nonzero sequence.

We now begin to characterize the closed range property of Rhaly operators.

Theorem 3.25. If a is a finite sequence, then Ra has a closed range, i.e.

Ra(H2(E, β)) = Ra(H2(E, β)).

Proof. Suppose a is a finite sequence. If a is the zero sequence, the resultfollows trivially. Otherwise, there exists an N ∈ N such that for all n > N,an = 0 and ak 6= 0 for some 1 ≤ k ≤ N. We define the set A as the set ofindices of an such that an 6= 0, i.e.

A := {n ∈ N : an 6= 0}.

Since a is a finite sequence and is not the zero sequence, A is a nonempty finiteset. Hence, we write A as

A = {n1, n2, ..., ni}

for some natural number 1 ≤ i ≤ N . Note that each nj ∈ N and we will alsoassume these numbers are in increasing order.

In order to show that Ra has a closed range, let (ym)∞m=1 be a sequence inRa(H

2(E, β)) such that ym → y in norm. Since ym ∈ Ra(H2(E, β)), there

exists xm ∈ H2(E, β) such that ym = Ra(xm) for each m. Our aim is to showthat y ∈ Ra(H

2(E, β)).

Since the sequences a and λn are already determined, the sequence terms ymare only determined by the coefficients of the Dirichlet series xm, i.e.

ym =N∑n=1

(an

n∑k=1

ck,m

)e−λnz

25

where xm =∑∞

k=1 ck,me−λkz ∈ H2(E, β). Since ym → y in norm, ym is Cauchy

in Ra(H2(E, β)). Let ε > 0 be given. Then there exists a K ∈ N such that for

any a, b ≥ K,

‖ya − yb‖ =N∑n=1

|an|2∣∣∣∣∣

n∑k=1

(ck,a − ck,b)

∣∣∣∣∣2

β2n

=∑n∈A

|an|2∣∣∣∣∣

n∑k=1

(ck,a − ck,b)

∣∣∣∣∣2

β2n < ε.

Claim: For all natural numbers 1 ≤ r ≤ i,(nr∑

k=nr−1+1

ck,m

)∞m=1

is Cauchy in C. (Here we define n0 = 0.) We do this by strong induction.When r = 1, we have

|an1 |2∣∣∣∣∣n1∑k=1

(ck,a − ck,b)

∣∣∣∣∣2

β2n1≤∑n∈A

|an|2∣∣∣∣∣

n∑k=1

(ck,a − ck,b)

∣∣∣∣∣2

β2n < ε.

Since an1 , βn1 6= 0, the claim is proven for r = 1. Suppose for some 1 ≤ j ≤ i−1,the claim is true for r = 1, ..., j. As anj+1

, βnj+16= 0, note that

|anj+1|2∣∣∣∣∣nj+1∑k=1

(ck,a − ck,b)

∣∣∣∣∣2

β2nj+1≤∑n∈A

|an|2∣∣∣∣∣

n∑k=1

(ck,a − ck,b)

∣∣∣∣∣2

β2n < ε,

so that

(∑nj+1

k=1 ck,m

)∞m=1

is Cauchy in C. Then

∣∣∣∣∣nj+1∑

k=nj+1

ck,m

∣∣∣∣∣ =

∣∣∣∣∣nj+1∑k=1

ck,m −n1∑k=1

ck,m − ...−nj∑

k=nj−1+1

ck,m

∣∣∣∣∣≤

∣∣∣∣∣nj+1∑k=1

ck,m

∣∣∣∣∣+

∣∣∣∣∣n1∑k=1

ck,m

∣∣∣∣∣+ ...+

∣∣∣∣∣nj∑

k=nj−1+1

ck,m

∣∣∣∣∣.By the induction hypothesis and

(∑nj+1

n=1 ck,m

)∞m=1

is Cauchy in C, we can see

that the claim is proven for r = j + 1. Hence the claim is true.

26

Since C is complete, for each 1 ≤ r ≤ i, there exists hnr ∈ C such that

nr∑k=nr−1+1

ck,m → hnr

as m→∞. By the earlier claim, since there are finitely many such sequences(nr∑

k=nr−1+1

ck,m

)∞m=1

,

we may choose a K1 ∈ N such that for any 1 ≤ r ≤ i and m ≥ K1,∣∣∣∣ nr∑k=nr−1+1

ck,m − hnr∣∣∣∣ < ε1/2√

2 ·N3/2 ·max1≤r≤i{|anr |} ·max1≤i≤N{βnr}.

Define a complex sequence (dk)∞k=1 such that

dk =

0, if k 6∈ A, k ≤ N

hnr , if k = nr for some r ∈ {1, ..., i}ck,1, if k > N.

Based on this sequence dk, define a new function

f(z) =N∑n=1

(an

n∑k=1

dk

)e−λnz.

We can see that f ∈ Ra(H2(E, β)) as

∞∑k=1

|dk|2β2k =

N∑k=1

|dk|2β2k +

∞∑k=N+1

|dk|2β2k

=i∑

r=1

|hnr |2β2k +

∞∑k=N+1

|ck,1|2β2k <∞,

as x1 =∑∞

k=1 ck,1e−λkz is in H2(E, β).

27

Note that if m ≥ K1, and any natural number 1 ≤ j ≤ i,∣∣∣∣∣nj∑k=1

ck,m −nj∑k=1

dk

∣∣∣∣∣ =

∣∣∣∣∣nj∑k=1

ck,m −i∑

r=1

hnr

∣∣∣∣∣=

∣∣∣∣∣j∑r=1

( nr∑k=nr−1+1

ck,m − hnr)∣∣∣∣∣

≤j∑r=1

∣∣∣∣ nr∑k=nr−1+1

ck,m − hnr∣∣∣∣

<

j∑r=1

ε1/2√2 ·N3/2 ·max1≤r≤i{|anr |} ·max1≤r≤i{βnr}

≤ ε1/2√2 ·N1/2 ·max1≤r≤i{|anr |} ·max1≤r≤i{βnr}

.

We now claim that ym → f in norm. For m ≥ K1, we have

‖ym − f‖ =

∥∥∥∥∥N∑n=1

(an

n∑k=1

ck,m

)e−λnz −

N∑n=1

(an

n∑k=1

dk

)e−λnz

∥∥∥∥∥=

i∑r=1

|anr |2∣∣∣∣∣nr∑k=1

ck,m −nr∑k=1

dk

∣∣∣∣∣2

β2nr

<ε

2·

i∑r=1

|anr |2

max1≤r≤i{|anr |}2·

β2nr

max1≤r≤i{βnr}21

N≤ ε

2.

Now, since ym → y in norm, there exists K2 ∈ N such that for all m ≥ K2,

‖ym − y‖ <ε

2.

Choosing M = max{K1, K2}, for all m ≥M, we have

‖y − f‖ ≤ ‖y − ym‖+ ‖ym − f‖ <ε

2+ε

2= ε,

and this implies that y ≡ f ∈ Ra(H2(E, β)) and the proof is complete.

Under the additional assumptions that Ra is bounded on the small weightedHilbert space of entire Dirichlet series, we can obtain a converse statement ofTheorem 3.25. Compactness is needed in this case, which in turn is true if thedomain of Ra is the small weighted Hilbert space of entire Dirichlet series (seeTheorem 3.19).

28

Theorem 3.26. Suppose Ra is bounded on the small weighted Hilbert space ofentire Dirichlet series. If Ra has a closed range, then a is a finite sequence.

Proof. Suppose Ra has a closed range. Since Ra is bounded, Theorem 3.19 tellsus that Ra is a compact on the small weighted Hilbert space of entire Dirichletseries and hence it must be finite-dimensional (see [10,Theorem 4.18b]). As-sume on the contrary that a is not a finite sequence. Choosing an orthonormalbasis {ek} as the orthonormal basis in Theorem 3.18, it follows that

dim(Ra(H2(E, β))) ≥ dim(span{Ra(ek) : k ∈ N}) =∞,

which is a contradiction. Therefore a is a finite sequence.

Remark 3.27. Let X, Y be Banach spaces. There is actually a well-knownapproach to show that a bounded operator T : X → Y has a closed range; aninjective operator T has a closed range if and only if it is bounded below, thatis, there exists an M > 0 such that for all x ∈ X, M ‖x‖ ≤ ‖T (x)‖ . However,notice that we are unable to use this theorem to show that Ra has a closedrange in Theorem 3.25. The reason is that if a is a finite sequence, then byTheorem 3.24, Ra is not injective.

Summarizing Theorems 3.25 and 3.26, we obtain a characterization for theclosed range property of bounded Rhaly operators.

Corollary 3.28. Suppose Ra is bounded on the small weighted Hilbert spaceof entire Dirichlet series. Then Ra has a closed range if and only if a is afinite sequence.

29

Chapter 4

Conclusion and Future Work

In this thesis, foundations were laid for Rhaly operators on the Hilbert space ofentire Dirichlet series and certain important properties like invariance, bound-edness were shown. Compactness, Hilbert-Schmidtness and Schatten p-classinclusions were proved on a smaller subset of the Hilbert space of entire func-tions, namely when the condition (3.5) is true.

We would also like to point out that the results in Theorem 3.19 are analogousto that of the results in [11], except the fact that the invariance and boundedproperties of Ra are independent of condition (3.5), which in [11], the bounded-ness and invariance properties are dependent on the condition (3.5). In otherwords, the boundedness and invariance properties hold for not just the smallweighted Hilbert space of entire Dirichlet series, but also the more generalspace H2(E, β).

That being said, there are possibilities yet to be explored; namely

• In Theorem 3.19, the compactness property is proven using Hilbert-Schmidtness of the operator Ra, which was in turn dependent on con-dition (3.5). However, not all compact operators are Hilbert-Schmidt,hence compactness of Ra may be independent of (3.5); is it true that Ra

is compact if (3.5) is not satisfied, i.e.

∞∑k=1

1

β2k

= +∞?

• Other properties of Ra can also be investigated, such as surjectivity,compact difference, cyclicity and pertubation, etc.

• What is the numerical range of Ra? (see [6, Section 2.3])

30

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