CUBO A Mathematical Journal
Vol.14, No¯ 03, (41–57). October 2012

Spectral results for operators commuting with translations
on Banach spaces of sequences on Zk and Z+

Violeta Petkova

Université Metz UFR MIM,

Laboratoire de Mathmatiques et Applications de Metz,

UMR 7122, Ile du Saulcy 57045 Metz Cedex 1, France

email: petkova@univ-metz.fr

ABSTRACT

We study the spectrum of multipliers (bounded operators commuting with the shift

operator S) on a Banach space E of sequences on Z. Given a multiplier M, we prove

that M̃(σ(S)) ⊂ σ(M) where M̃ is the symbol of M. We obtain a similar result for the
spectrum of an operator commuting with the shift on a Banach space of sequences on

Z+. We generalize the results for multipliers on Banach spaces of sequences on Zk.

RESUMEN

Estudiamos el espectro de los multiplicadores (operadores acotados que conmutan con el

operador shift S) en un espacio de Banach E de sucesiones en Z. Dado un multiplicador

M, probamos que M̃(σ(S)) ⊂ σ(M) donde M̃ es el śımbolo de M. Obtenemos un
resultados similar para el espectro de un operador que conmuta con el shift en un espacio

de Banach de sucesiones en Z+. Generalizamos los resultados sobre multiplicadores en

espacios de Banach de sucesiones en Zk.

Keywords and Phrases: Multiplier, Toeplitz operator, shift operator, space of sequences, spec-

trum of multiplier, joint spectrum of translations.

2010 AMS Mathematics Subject Classification: Primary 47B37, 47B35; Secondary 47A10.



42 Violeta Petkova CUBO
14, 3 (2012)

1 Introduction

Let E ⊂ CZ be a Banach space of complex sequences (x(n))n∈Z. Denote by S : CZ −→ CZ, the shift
operator defined by Sx = (x(n − 1))n∈Z, for x = (x(n))n∈Z ∈ CZ, so that S−1x = (x(n + 1))n∈Z.
Let F(Z) be the set of sequences on Z, which have a finite number of non-zero elements and

assume that F(Z) ⊂ E. We will call a multiplier on E every bounded operator M on E such
that MSa = SMa, for every a ∈ F(Z). Denote by µ(E) the space of multipliers on E. For
z ∈ T = {z ∈ C : |z| = 1}, consider the map E ∋ x −→ ψz(x) given by ψz(x) = (x(n)zn)n∈Z.
Notice that if we assume that ψz(E) ⊂ E for all z ∈ T and if for all n ∈ Z, the map

pn : E ∋ x −→ x(n) ∈ C

is continuous, then from the closed graph theorem it follows that the map ψz is bounded on E. In

this paper, we deal with Banach spaces of sequences on Z satisfying only the following three very

natural hypothesis:

(H1) The set F(Z) is dense in E.

(H2) For every n ∈ Z, pn is continuous from E into C.
(H3) We have ψz(E) ⊂ E, ∀z ∈ T and supz∈T ‖ψz‖ < +∞.
We give some examples of spaces satisfying our hypothesis.

Example 1. Let ω be a weight (a sequence of positive real numbers) on Z. Set

lpω(Z) =
{
(x(n))n∈Z ∈ CZ;

∑

n∈Z

|x(n)|pω(n)p < +∞
}
, 1 ≤ p < +∞

and ‖x‖ω,p =
(∑

n∈Z
|x(n)|pω(n)p

) 1
p

. It is easy to see that the Banach space l
p
ω(Z) satisfies our

hypothesis. Moreover, the operator S (resp. S−1) is bounded on l
p
ω(Z) if and only if

sup
n∈Z

ω(n + 1)

ω(n)
< +∞

(
resp. sup

n∈Z

ω(n − 1)

ω(n)
< +∞

)
.

Example 2. Let K be a convex, non-decreasing, continuous function on R+ such that K(0) = 0
and K(x) > 0, for x > 0. For example, K may be xp, for 1 ≤ p < +∞ or xp+sin(log(− log(x))), for
p > 1 +

√
2. Let ω be a weight on Z. Set

lK,ω(Z) =
{
(x(n))n∈Z ∈ CZ;

∑

n∈Z

K
(
|x(n)|

t

)
ω(n) < +∞, for some t > 0

}

and ‖x‖ = inf
{
t > 0,

∑
n∈Z

K
(

|x(n)|

t

)
ω(n) ≤ 1

}
. The space lK,ω(Z), called a weighted Orlicz

space (see [2], [3]), is a Banach space satisfying our hypothesis.

Example 3. Let (q(n))n∈Z be a real sequence such that q(n) ≥ 1, for all n ∈ Z. For
a = (a(n))n∈Z ∈ CZ, set ‖a‖{q} = inf

{
t > 0,

∑
n∈Z

∣∣∣a(n)t
∣∣∣
q(n)

≤ 1
}
. Consider the space

l{q} = {a ∈ CZ; ‖a‖{q} < +∞}, which is a Banach space (see [1]) satisfying our hypothesis.
Notice that if limn→+∞ |q(n + 1) − q(n)| 6= 0 and if supn∈Z q(n) < +∞, then either S or S−1 is



CUBO
14, 3 (2012)

Spectral results for operators commuting with translations ... 43

not bounded (see [4]).

It is easy to see that if S(E) ⊂ E, then by the closed graph theorem the restriction S|E of S to E
is bounded from E into E. From now on we will say that S (resp. S−1) is bounded when S(E) ⊂ E
(resp. S−1(E) ⊂ E). If S(E) ⊂ E, we will call σ(S) the spectrum of the operator S with domain
E. If S is not bounded, denote by σ(S) (resp. ρ(S)) the spectrum (resp. the spectral radius) of S,

where S is the smallest extension of S|F(Z) as a closed operator. Recall that the domain D(S) of S

is given by

D(S) = {x ∈ E, ∃(xn)n∈N ⊂ F(Z) s.t. xn −→ x andSxn −→ y ∈ E}

and for x ∈ D(S) we set Sx = y. We will denote by
◦

A (resp. δ(A)) the interior (resp. the boundary)

of the set A. Denote by ek the sequence such that ek(n) = 0 (resp. 1), if n 6= k (resp. n = k).
For a multiplier M, we set M̂ = M(e0) and it is easy to see that

Ma = M̂ ∗ a, ∀a ∈ F(Z). (1)

Given a ∈ CZ, define ã(z) =
∑
n∈Z

a(n)zn and notice that if a ∈ l2(Z), then ã ∈ L2(T).
Denote by M̃ the function

z −→ M̃(z) =
∑

n∈Z

M̂(n)zn.

Usually, M̃ is called the symbol of M. It is easy to see that on the space of formal Laurent series

we have the equality

M̃a(z) = M̃(z)ã(z), ∀z ∈ C, ∀a ∈ F(Z). (2)

However, it is difficult to determine for which z ∈ C the series M̃(z) converges. For r > 0, let Cr
be the circle of center 0 and radius r. Recall the following result established in [8] (Theorem 1).

Theorem 1. 1) If S is not bounded, but S−1 is bounded, then ρ(S) = +∞ and if S is bounded, but

S−1 is not bounded, then ρ(S−1) = +∞.

2) We have σ(S) =
{
z ∈ C, 1

ρ(S−1)
≤ |z| ≤ ρ(S)

}
.

3) Let M ∈ µ(E). For r > 0 such that Cr ⊂ σ(S), we have M̃ ∈ L∞(Cr) and |M̃(z)| ≤ ‖M‖, a.e.
on Cr.

4) If ρ(S) > 1
ρ(S−1)

, then M̃ is holomorphic on
◦

σ(S).

If S and S−1 are bounded, denote by IE the interval
[

1
ρ(S−1)

,ρ(S)
]
. If S (resp. S−1) is not

bounded, denote by IE the interval
[

1
ρ(S−1)

,+∞
[
(resp. ]0,ρ(S)]).

The purpose of this paper is to use the symbol of an operator M ∈ µ(E) in order to characterize
its spectrum. We deal with three different setups. First we study the multipliers on E, next

we examine Toeplitz operators on a Banach space of sequences on Z+ and finally we deal with

multipliers on a Banach space of sequences on Zk. For φ ∈ F(Z), we denote by Mφ the operator



44 Violeta Petkova CUBO
14, 3 (2012)

of convolution by φ on E. Let S be the closure with respect to the operator norm topology of the

algebra generated by the operators Mφ, for φ ∈ F(Z). Our first result is

Theorem 2. 1) If M ∈ µ(E), we have M̃(σ(S)) ⊂ σ(M).
2) If M ∈ S , then σ(M) \ {0} ⊂ M̃(σ(S)) ⊂ σ(M).

Notice that here for a set A, we denote by A the closure of A. If M ∈ µ(E), then M̃(σ(S))
denotes the essential range of M̃ on σ(S). Notice that M̃ is holomorphic on

◦

σ(S) and essentially

bounded on the boundary of σ(S). In general we have not spectral calculus for the operators in

µ(E) and it seems difficult to characterize the spectrum of M ∈ µ(E) without using its symbol.
We also study a similar spectral problem for Toeplitz operators. Let E ⊂ CZ+ be a Banach

space and let F(Z+) (resp. F(Z−)) be the space of the sequences on Z+ (resp. Z−) which have

a finite number of non-zero elements. By convention, we will say that x ∈ F(Z) is a sequence of
F(Z+) (resp. F(Z−)) if x(n) = 0, for n < 0 (resp. n > 0). We will assume that E satisfies the

following hypothesis:

(H1) The set F(Z+) is dense in E.
(H2) For every n ∈ Z+, the application pn : x −→ x(n) is continuous from E into C.
(H3) For x = (x(n))n∈Z+ ∈ E, we have γz(x) = (znx(n))n∈Z+ ∈ E, for every z ∈ T and
supz∈T ‖γz‖ < +∞.

Definition 1. We define on CZ
+

the operators S1 and S−1 as follows.

Foru ∈ CZ
+

, (S1(u))(n) = 0, ifn = 0and (S1(u))(n) = u(n − 1), ifn ≥ 1

(S−1(u))(n) = u(n + 1), forn ≥ 0.

For simplicity, we note S instead of S1. It is easy to see that if S(E) ⊂ E, then by the closed
graph theorem the restriction S|E of S to E is bounded from E into E. We will say that S (resp.

S−1) is bounded when S(E) ⊂ E (resp S−1(E) ⊂ E). Next, if S|E (resp. S−1|E) is bounded, σ(S)
(resp. σ(S−1)) denotes the spectrum of S|E (resp. S−1|E). If S (resp. S−1) is not bounded, σ(S)

(resp. σ(S−1)) denotes the spectrum of the smallest closed extension of S|F(Z+) (resp. S−1|F(Z+)).

For u ∈ l2(Z−) ⊕ E introduce

(P+(u))(n) = u(n), ∀n ≥ 0 and (P+(u))(n) = 0, ∀n < 0.

If S1 and S−1 are the shift and the backward shift on l
2(Z−)⊕E, then S = P+S1 and S−1 = P+S−1.

Example 4. Let w be a positive sequence on Z+. Set

lpw(Z
+) =

{
(x(n))n∈Z+ ∈ CZ

+

;
∑

n∈Z+

|x(n)|pw(n)p < +∞
}
, 1 ≤ p < +∞



CUBO
14, 3 (2012)

Spectral results for operators commuting with translations ... 45

and ‖x‖w,p =
(∑

n∈Z+
|x(n)|pw(n)p

) 1
p

. It is easy to see that the Banach space l
p
w(Z

+) satisfies

our hypothesis. The operator S (resp. S−1) is bounded on l
p
w(Z

+), if and only if, w satisfies

sup
n∈Z+

w(n + 1)

w(n)
< +∞

(
resp. sup

n∈Z+

w(n)

w(n + 1)
< +∞

)
.

Definition 2. A bounded operator T on E is called a Toeplitz operator, if we have:

(S−1TS)u = Tu, ∀u ∈ F(Z+).

Denote by TE the space of Toeplitz operators on E.

It is easy to see that if T commutes either with S or with S−1, then T is a Toeplitz operator.

Indeed, if TS−1 = S−1T, then T = S−1TS. Notice that if T ∈ TE, we have Tu = P+S−nTSnu, for
all u ∈ F(Z+) and all n > 1. Here Sn (resp. S−n) denotes (S1)n (resp. (S−1)n) where S1 (resp.
S−1) is the shift (resp. the backward shift) on l

2(Z−) ⊕ E.

Remark that we have S−1S = I, however SS−1 6= I and this is the main difficulty in the
analysis of Toeplitz operators.

Given a Toeplitz operator T, set T̂(n) = (Te0)(n) and T̂(−n) = (Ten)(0), for n ≥ 0 and define
T̂ = (T̂(n))n∈Z. It is easy to see that we have

Tu = P+(T̂ ∗ u), ∀u ∈ F(Z+). (3)

Set T̃(z) =
∑
n∈Z

T̂(n)zn, for z ∈ C. Notice that the series T̃(z) could diverge.
If S and S−1 are bounded, we will denote by IE the interval

[
1

ρ(S−1)
,ρ(S)

]
. If S (resp. S−1) is

not bounded, then IE denotes
[

1
ρ(S−1)

,+∞
[ (

resp.
]
0,ρ(S)

])
.

If S and S−1 are bounded, denote by UE the set
{
z ∈ C, 1

ρ(S−1)
≤ |z| ≤ ρ(S)

}
. If S (resp. S−1)

is not bounded then UE denotes
{
z ∈ C, 1

ρ(S−1)
≤ |z|

} (
resp.

{
z ∈ C, |z| ≤ ρ(S)

})
. We have

the following result (see Theorem 2 in [8])

Theorem 3. Let T be a Toeplitz operator on E.

1) For r ∈
[

1
ρ(S−1)

,ρ(S)
]
, if ρ(S) < +∞ or for r ∈

[
1

ρ(S−1)
,+∞

[
, if ρ(S) = +∞ we have

T̃ ∈ L∞(Cr) and |T̃(z)| ≤ ‖T‖, a.e. on Cr.
2) If

◦

UE is not empty, T̃ ∈ H∞(
◦

UE), where H∞(
◦

UE) is the space of holomorphic and essentially

bounded functions on
◦

UE.

Denote by µ(E) the set of bounded operators on E commuting with either S or S−1. As

mentioned above µ(E) ⊂ TE. It is clear that the operators (Sn)n≥0 and ((S−1)n)n≥0 are included
in µ(E). In this paper we prove the following



46 Violeta Petkova CUBO
14, 3 (2012)

Theorem 4. If S and S−1 are bounded operators, we have

σ(S) = {z ∈ C : |z| ≤ ρ(S)}. (4)

σ(S−1) = {z ∈ C : |z| ≤ ρ(S−1)}. (5)

For the right R and left L weighted shifts on l2(N) the results (4), (5) are classical (see for

instance, [11]). Moreover, it is well known that the spectrum of R and L have a circular symmetry

([12]). The proofs of these results for R and L use the structure of l2(Z+) and the analysis of

the point spectrum is given by a direct calculus. In the general situation we deal with such an

approach is not possible and our results on the symbols of Toeplitz operators play a crucial role.

First we establish in Proposition 1 the relation { 1
ρ(S−1)

≤ |z| ≤ ρ(S)} ⊂ σ(S) and next we obtain
(4). It seems that Theorem 4 is the first result concerning the description of σ(S) and σ(S−1) in

the general setup when (H1)- (H3) hold.

For operators commuting either with S or S−1 we have the following

Theorem 5. Suppose that S and S−1 are bounded. Let T be a bounded operator on E commuting

with S. Then we have

T̃(
◦

σ(S)) ⊂ σ(T). (6)

If T is a bounded operator on E commuting with S−1, we have

T̃(
◦

σ(S−1)) ⊂ σ(T). (7)

For φ ∈ CZ, define
Tφf = P

+(φ ∗ f), ∀f ∈ E.

If φ ∈ CZ is such that Tφ is bounded on E (it is the case if for example φ ∈ F(Z+)), then
Tφ ∈ µ(E). The author has established similar results for multipliers and Winer-Hopf operators
in weighted spaces L2ω(R) and L

2
w(R

+) (see [6], [9]). The spaces considered in this paper are much

more general then weighted l2ω(Z) and l
2
ω(Z

+) spaces. Here we consider not only Hilbert spaces,

but also Banach spaces which may have a complicated structure (see Example 2 and Example

3). Moreover, we study multipliers on spaces where the shift is not a bounded operator. In these

general cases our spectral results are based heavily on the symbolic representation and this was the

main motivation for proving the existence of symbols for the operators of the classes we consider.

For ψ ∈ Cc(R+), denote by Tψ the operator defined on Lpω(R+) by

(Tψf)(x) = P+(ψ ∗ f)(x), a.e.

Set β0 = limt→+∞ ln ‖St‖
1
t . A recent result of the author (see [10]) shows that if Tψ commutes

with (St)t≥0 then

σ(Tψ) = ψ̂(V),



CUBO
14, 3 (2012)

Spectral results for operators commuting with translations ... 47

where V = {z ∈ C, Imz ≤ β0}. It is natural to conjecture that

σ(Tφ) = φ̂(UE)

for Tφ with φ ∈ F(Z) commuting with S.

In Section 4, we study the so called joint spectrum for translation operators on a Banach space

of sequences on Zk and we generalize the results of Section 2. In Theorem 7 we prove that the

spectrum of a multiplier (bounded operator commuting with the translations) on a very general

Banach space E of sequences on Zk is related to the image under its symbol of the joint spectrum
of the translations S1, ...,Sk (see Section 4 for the definitions). This joint spectrum denoted by
σA(S1, ...,Sk) (see Section 4) is very important in our analysis. Notice that σA(S1, ...,Sk) ⊂
σ(S1) × ...×σ(Sk) but in general the inclusion is strict. The fact that the symbol of a multiplier is
holomorphic on the interior of σA(S1, ...,Sk) plays a crucial role. To our best knowledge it seems
that Theorem 7 is the first result in the literature concerning the spectrum of operators commuting

with translations on a Banach space of sequences on Zk.

2 Spectrum of a multiplier

First, consider the case of the multipliers on a Banach space E satisfying (H1)-(H3) and suppose

that S or S−1 is bounded on E. Define {z ∈ C, |z| ∈ IE}.. To prove Theorem 2, we will need the
following lemma established in [8] (Lemma 4).

Lemma 1. For φ ∈ F(Z), we have |M̃φ(z)| ≤ ‖Mφ‖, ∀z ∈ UE.

Definition 3. For a ∈ CZ, and r ∈ R, define the sequence (a)r so that

(a)r(n) = a(n)r
n, ∀n ∈ Z.

Lemma 2. Let r ∈ IE and f ∈ E be such that (f)r ∈ l2(Z). If M ∈ µ(E), we have

(Mf)r = (M̂r ∗ (f)r), (M̃f)r(z) = M̃(rz)(̃f)r(z), ∀z ∈ T

and (̃Mf)r ∈ L2(T).

Lemma 2 is a generalization of (1).

Proof. The proof uses the arguments exposed in [8] with some modifications. For the

completeness we give here the details. Let M ∈ µ(E). Let (Mk)k∈N be a sequence such that
limk→+∞ ‖Mkx − Mx‖ = 0, ∀x ∈ E, ‖Mk‖ ≤ ‖M‖ and Mk = Mφk, where φk ∈ F(Z), ∀k ∈ N.
The existence of this sequence is established in [8] (Lemma 3). Let r ∈ IE. We have |(̃φk)r(z)| ≤
‖Mφk‖ ≤ ‖M‖, ∀z ∈ T, ∀k ∈ N. We can extract from

(
(̃φk)r

)
k∈N

a subsequence which converges



48 Violeta Petkova CUBO
14, 3 (2012)

with respect to the weak topology σ(L∞(T),L1(T)) to a function νr ∈ L∞(T). For simplicity, this
subsequence will be denoted also by

(
(̃φk)r

)
k∈N

. We obtain

lim
k→+∞

∫

T

(
(̃φk)r(z)g(z) − νr(z)g(z)

)
dz = 0, ∀g ∈ L1(T)

and ‖νr‖∞ ≤ ‖M‖. Fix f ∈ E such that (f)r ∈ l2(Z). It is clear that

lim
k→+∞

∫

T

(
(̃φk)r(z)(̃f)r(z)g(z) − νr(z)(̃f)r(z)g(z)

)
dz = 0, ∀g ∈ L2(T).

We observe that the sequence
(
(̃φk)r(̃f)r

)
k∈N

converges with respect to the weak topology

of L2(T) to νr(̃f)r. Set ν̂r(n) =
1
2π

∫π
−π
νr(e

it)e−itndt, for n ∈ Z and let ν̂r = (ν̂r(n))n∈Z be the
sequence of the Fourier coefficients of νr. The Fourier transform from l

2(Z) to L2(T) defined by F :
l2(Z) ∋ (f(n))n∈Z −→ f̃|T ∈ L2(T) is unitary, so the sequence

(
(Mφkf)r

)
k∈N

=
(
(φk)r ∗ (f)r

)
k∈N

converges to ν̂r ∗ (f)r with respect to the weak topology of l2(Z). Taking into account that E
satisfies (H2), for n ∈ Z we obtain

lim
k→+∞

|((Mφkf)r − (Mf)r)(n)| ≤ lim
k→+∞

C‖Mφkf − Mf‖ = 0.

Thus we deduce that

(Mf)r(n) = (ν̂r ∗ (f)r)(n), ∀n ∈ Z, ∀f ∈ E,

such that (f)r ∈ l2(Z). This implies (M̂)r ∗ (f)r = ν̂r ∗ (f)r,∀f ∈ F(Z) and then we get (M̂)r = ν̂r.
We conclude that (Mf)r = (M̂)r ∗ (f)r, ∀f ∈ E such that (f)r ∈ l2(Z) and then we have

(M̃f)r(z) = M̃(rz)(̃f)r(z), ∀z ∈ T.

Since (̃f)r ∈ L2(T) and M̃ ∈ L∞(UE), it is clear that (̃Mf)r ∈ L2(T). ✷

Proof of Theorem 2. Let M ∈ µ(E). Suppose that α /∈ σ(M). Then we have

(M − αI)−1 ∈ µ(E).

For z ∈ σ(S), we obtain
(

˜(M − αI)−1f
)
(z) =

( ∑

n∈Z

̂(M − αI)−1(n)zn
)( ∑

n∈Z

f(n)zn
)
,

for all f ∈ E, such that for all r ∈ IE, (f)r ∈ l2(Z). If g ∈ F(Z), following Lemma 2, we may replace
f by (M − αI)g. We get

g̃(z) =
( ∑

n∈Z

̂(M − αI)−1(n)zn
)( ∑

n∈Z

((M − αI)g)(n)zn
)



CUBO
14, 3 (2012)

Spectral results for operators commuting with translations ... 49

= ˜(M − αI)−1(z)(M̃g(z) − αg̃(z)) = ˜(M − αI)−1(z)(M̃(z) − α)g̃(z), ∀g ∈ F(Z),

for all z ∈ σ(S). This implies that for fixed r ∈ IE,

˜(M − αI)−1(rη)(M̃(rη) − α) = 1, ∀η ∈ T.

Since, ˜(M − αI)−1 is holomorphic on
◦

σ(S) and essentially bounded on δ(σ(S)) (see Theorem 1),

we obtain that M̃(z) 6= α, for every z ∈
◦

σ(S) and for almost every z ∈ δ(σ(S)). We conclude that
M̃(σ(S)) ⊂ σ(M), which proves the first part of the theorem. For te second one, let M ∈ S. Then
there exists a sequence (Mφn) with φn ∈ F(Z) such that limn→+∞ ‖Mφn − M‖ = 0. Notice that
from Lemma 1 it follows that

|M̃φn(z)| ≤ ‖Mφn‖ ≤ ‖M‖, ∀z ∈ UE.

Taking into account that |M̃φn(z)−M̃φk(z)| ≤ ‖Mφn −Mφk‖, ∀z ∈ UE, and the fact that (Mφn)
converges with respect to the norm operator theory, we conclude that (M̃φn) converges uniformly

on UE to a function µM. We observe that (M̃φn)r converges to µM(r.) with respect to the weak

topology σ(L∞(T),L1(T)). So we can identify M̃(rz) and µM(rz) for z ∈ T.
Consequently, M̃ is continuous on δ(UE). Let λ ∈ σ(M) \ {0}. Then there exists a character

γ on S such that λ = γ(M). For k ∈ N∗, denote by Sk the operator (S1)k. We have

γ(Mφn) = γ(
∑

k∈Z

φ̂n(k)Sk) =
∑

k∈Z

φ̂n(k)γ(S)
k

and we get γ(M) = limn→+∞ γ(Mφn) = limn→+∞ M̃φn(γ(S)) = M̃(γ(S)). We conclude that

σ(M) \ {0} ⊂ M̃(σ(S)). ✷
Now suppose that M /∈ S. If α ∈ σ(M) \ {0}, then α = γ(M), where γ is a character on the

commutative Banach algebra µ(E). Following [8] (Lemma 3), there exists a sequence (Mφn), with

φn ∈ F(Z) such that limn→+∞ ‖Mφna−Ma‖ = 0, ∀a ∈ E and we have limn→+∞ M̃φn(z) = M̃(z),
for all z ∈

◦

σ(S) and for almost every z ∈ δ(σ(S)). If we suppose that γ(S) ∈
◦

σ(S) we have

lim
n→+∞

γ(Mφn) = lim
n→+∞

M̃φn(γ(S)) = M̃(γ(S)),

but in the general case we do not have

lim
n→+∞

γ(Mφn) = γ(M),

because (Mφn) converges to M with respect to the strong operator theory and may be not for the

norm operator topology.



50 Violeta Petkova CUBO
14, 3 (2012)

3 Spectrum of an operator commuting either with S or S−1
on E

In this section we consider a Banach space E satisfying the conditions (H1) − (H3). Suppose that
S and S−1 are bounded on E.

Notice that it is easy to see that, for φ ∈ F(Z), if Tφ commutes with S (resp. S−1) then
φ ∈ F(Z+) (resp. F(Z−)).

Lemma 3. For T ∈ TE, r ∈ IE and for a ∈ E such that (a)r ∈ l2(Z+) we have

(Ta)r = P
+((T̂)r ∗ (a)r) (8)

and then (Ta)r ∈ l2(Z+).

Lemma 3 is a generalization of the property (3).

Proof. Let T be a bounded operator in TE and let (φk)k∈N ⊂ F(Z) be such that

lim
k→+∞

‖Tφka − Ta‖ = 0, ∀a ∈ E

and ‖Tφk‖ ≤ ‖T‖, ∀k ∈ N. The existence of the sequence (Tφk) is established in [8] (Lemma 5).
Fix r ∈ IE. We have (see Lemma 6 in [8]), |(̃φk)r(z)| ≤ ‖Tφk‖ ≤ ‖T‖, ∀z ∈ T, ∀k ∈ N. We
can extract from

(
(̃φk)r

)
k∈N

a subsequence which converges with respect to the weak topology

σ(L∞(T),L1(T)) to a function νr ∈ L∞(T). For simplicity, this subsequence will be denoted also by(
(̃φk)r

)
k∈N

. Let a ∈ E be such that (a)r ∈ l2(Z+). We conclude that,
(
(̃φk)r(̃a)r

)
k∈N

converges

with respect to the weak topology of L2(T) to νr(̃a)r. Denote by ν̂r = (ν̂r(n))n∈Z the sequence of

the Fourier coefficients of νr. Since the Fourier transform from l
2(Z) to L2(T) is an isometry, the

sequence (φk)r ∗ (a)r converges to ν̂r ∗ (a)r with respect to the weak topology of l2(Z). On the
other hand,

(
Tφka

)
k∈N

converges to Ta with respect to the topology of E. Consequently, since E

satisfies (H2) we have

lim
k→+∞

|((Tφka)r − (Ta)r)(n)| ≤ lim
k→+∞

C‖Tφka − Ta‖ = 0, ∀n ∈ N.

We conclude that

(Ta)r = P
+(ν̂r ∗ (a)r), ∀a ∈ E such that (a)r ∈ l2(Z+). (9)

Since

(Ta)r = P
+
((T̂ ∗ a)r), ∀a ∈ F(Z+),

it follows that T̂(n)rn = ν̂r(n), ∀n ∈ Z. Then (9) implies obviously (8). Combining (8) with the
fact that T̂ ∈ l∞(Z), it is clear that if (a)r ∈ l2(Z+), then (Ta)r ∈ l2(Z+). ✷



CUBO
14, 3 (2012)

Spectral results for operators commuting with translations ... 51

For the proof of Theorem 4 we need the following

Proposition 1. Let T be a bounded operator in µ(E). Then we have

T̃(
◦

UE) ⊂ σ(T).

Proof. Let T ∈ µ(E) and suppose that λ /∈ σ(T). First we will show that (T − λI)−1 ∈ TE.
If TS = ST, then (T − λI)S = S(T − λI) and we obtain (T − λI)−1S = S(T − λI)−1. As we have

mentioned above this implies that (T −λI)−1 is a Toeplitz operator. In the same way we treat the

case when TS−1 = S−1T.

Set h(n) = ̂(T − λI)−1(n) and fix r ∈ IE. For all g ∈ E such that (g)r ∈ l2(Z+), applying Lemma
3 with (T −λI) ∈ TE and a = g we get (T −λI)g ∈ l2(Z+). Then allpying a second time Lemma 3
with (T − λI)−1 ∈ TE and a = (T − λI)g, we get

(g)r = P
+
(
(h)r ∗ ((T − λI)g)r

)
, ∀g ∈ E such that (g)r ∈ l2(Z+).

Since (h)r and ((T − λI)g)r are in l
2(Z+) (see Lemma 3), we have

‖(̃g)r‖L2(T) = ‖(g)r‖l2(Z+) = ‖P+((h)r ∗ ((T − λI)g)r)‖l2(Z+)

≤ ‖P+‖‖(h)r ∗ ((T − λI)g)r)‖l2(Z+) = ‖P+‖‖(̃h)r(T̃ − λ)(̃g)r‖L2(T)

≤ ‖P+‖‖(̃h)r‖L∞(T)‖(T̃ − λ)(̃g)r‖L2(T)

≤ C‖(T̃ − λ)(̃g)r‖L2(T), ∀g ∈ E such that (g)r ∈ l2(Z+). (10)

First suppose that 1 ∈
◦

IE. Then for r = 1, we get

‖g̃‖L2(T) ≤ C‖(T̃ − λ)g̃‖L2(T), ∀g ∈ E ∩ l2(Z+).

Assume that λ = T̃(z0) for z0 ∈ T ⊂
◦

UE. According to Theorem 3, T̃ is continuous on T and it is

easy to choose f ∈ L2(T) so that

2C‖(T̃ − λ)f‖L2(T) < ‖f‖L2(T). (11)

In fact, if |T̃(z) − λ| ≤ δ for |z − z0| < η(δ), we take f such that f(z) = 0 for z s.t. |z − z0| ≥ η(δ)
and ‖f‖L2(T) = 1. For δ > 0 such that 2Cδ < 1 we get the inequality (11). Let g ∈ l2(Z) be such
that f = g̃ and let β = C‖T̃ − λ‖∞. Fix ǫ > 0 so that ‖g̃‖L2(T) > (2β + 2)ǫ. Next let gǫ ∈ F(Z) be
such that ‖gǫ − g‖l2(Z) ≤ ǫ. Then we have

C‖(T̃ − λ)g̃ǫ‖L2(T) ≤ C‖(T̃ − λ)(g̃ǫ − g̃)‖L2(T) + C‖(T̃ − λ)g̃‖L2(T)

≤ βǫ + 1
2
‖g̃‖L2(T) < βǫ +

1

2
‖g̃ − g̃ǫ‖L2(T) +

1

2
‖g̃ǫ‖L2(T) < (β +

1

2
)ǫ +

1

2
‖g̃ǫ‖L2(T).



52 Violeta Petkova CUBO
14, 3 (2012)

On the other hand, ‖g̃ǫ‖L2(T) ≥ ‖g̃‖L2(T) −ǫ ≥ 2β+ǫ, hence (β+ 12)ǫ ≤
1
2
‖g̃ǫ‖L2(T). This implies

C‖(T̃ − λ)g̃ǫ‖L2(T) < ‖g̃ǫ‖L2(T).

Notice that for f ∈ l2(Z) and n ∈ Z+, we have S̃nf(z) = znf̃(z), ∀z ∈ T. Set h = SNgǫ, where
N ∈ Z+ is chosen so that SNgǫ ∈ F(Z+). We have

C‖(T̃ − λ)h̃‖L2(T) = C‖(T̃ − λ)S̃Ngǫ‖L2(T)

= C‖(T̃ − λ)g̃ǫ‖L2(T) < ‖g̃ǫ‖L2(T) = ‖h̃‖L2(T).
Taking into account (10), we obtain a contradiction and then T̃(z) ∈ σ(T) for z ∈ T.
Now let r ∈

◦

IE and r 6= 1. Repeating the above argument, we choose g ∈ F(Z+) so that

C‖(T̃ − λ)g̃‖L2(T) < ‖g̃‖L2(T).

Let h be the sequence defined by h(n) = g(n)r−n, ∀n ∈ Z+. Then g = (h)r and h ∈ F(Z+). We
have

C‖(T̃ − λ)(̃h)r‖L2(T) < ‖(̃h)r‖L2(T).
By using (10) once more, we obtain a contradiction and then T̃(Cr) ⊂ σ(T), where Cr is the circle
of center 0 and radius r and this completes the proof of the theorem. ✷

Proof of Theorem 4. The symbol of S is z −→ z and according to Proposition 1, we have
UE ⊂ σ(S). It remains to show that {z ∈ C, |z| < 1ρ(S−1)} ⊂ σ(S). We apply the argument of [9].
For 0 < |z| < 1

ρ(S−1)
we write

S−1 −
1

z
I = −

1

z

(
S−1(S − z)I

)
(12)

If z /∈ σ(S), then there exists g 6= 0 such that (S − z)g = e0. This implies (S−1 − 1z)g = 0 and we
obtain a contradiction with the fact that 1

|z|
> ρ(S−1). This completes the proof of (4).

Now we pass to the analysis of σ(S−1). As above assume that S−1 is bounded. Following [9],

we show first that for the approximative spectrum Π(S) of S we have

Π(S) ⊂ {z ∈ C, 1
ρ(S−1)

≤ |z| ≤ ρ(S)}.

In fact, for z 6= 0, if there exists a sequence fn, ‖fn‖ = 1 such that (S − z)fn → 0 as n → ∞, then
from (12) we deduce that (S−1 −

1
z
)fn → 0 and this yields

1
|z|

≤ ρ(S−1). On the other hand, if
0 ∈ Π(S), there exists a sequence fn, ‖fn‖ = 1 such that Sfn → 0 and this yields a contradiction
with the equality fn = S−1Sfn.

For the proof of (5) we use for z 6= 0 the adjoint operators S∗, S∗−1 and the equality

z
(
1

z
I − S∗

)
= S∗

(
(S−1)

∗ − zI
)
.



CUBO
14, 3 (2012)

Spectral results for operators commuting with translations ... 53

The symbol of S−1 is z −→ 1z and an application of Proposition 1 yields {
1
ρ(S)

≤ |z| ≤ ρ(S−1)} ⊂
σ(S−1). Next assume that 0 < |z| <

1
ρ(S)

. We are going to repeat the argument of the proof

of Theorem 3 in [9] and for completeness we give the proof. First, 0 ∈ σr(S), σr(S) being the
residual spectrum of S. In fact, if this is not true, 0 will be in Π(S) and this is a contradiction.

Secondly, we deduce that 0 will be an eigenvalue of the adjoint operator S∗. Let S∗g = 0 with

g 6= 0. If (S−1)∗ − zI is surjective, than there exists f 6= 0 such that ((S−1)∗ − z)f = g and we get
(1
z
− S∗)f = 0, hence 1

|z|
≤ ρ(S∗) = ρ(S) which is impossible. Thus z ∈ σ(S∗−1) and, passing to the

adjoint, we complete the proof. ✷.

For the proof of Theorem 5 we need the following

Lemma 4. Let φ ∈ F(Z+) (resp. F(Z−)). Then for z ∈ σ(S) (resp. z ∈ σ(S−1)), we have

|(̃φ)(z)| ≤ ‖Tφ‖ ≤ ‖T‖.

Proof. Suppose that |z| = ρ(S). Then z is in Π(S) and there exists a sequence (fn)n∈N ⊂ E
such that ‖fn‖ = 1 and limn→+∞ ‖Sfn − zfn‖ = 0. Then for φ ∈ F(Z+), we have for some N > 0,

‖φ ∗ fn − φ̃(z)fn‖ ≤
N∑

k=0

( sup
|k|≤N

|φ(k)|)‖Skfn − zkfn‖

and we obtain

lim
n→+∞

‖φ ∗ fn − φ̃(z)fn‖ = 0.

Since

|φ̃(z)| = ‖φ̃(z)fn‖ ≤ ‖φ̃(z)fn − φ ∗ fn‖ + ‖Tφfn‖,

it follows that |φ̃(z)| ≤ ‖Tφ‖. By using the maximum principle for the analytic function φ̃ we
complete the proof for z ∈ σ(S). For σ(S−1) we apply the same argument. ✷.

Lemma 5. Let T be a bounded operator on E commuting with S. Let r be such that there exists

z ∈ σ(S) with r = |z|. Then for a ∈ E such that (a)r ∈ l2(Z+) we have

(Ta)r = P
+((T̂)r ∗ (a)r) (13)

and then (Ta)r ∈ l2(Z+).

Proof. For the proof we apply Lemma 4 and the same arguments as those in the proof of

Lemma 3. ✷

By using Lemmas 4-5 and repeating the arguments of the proof of Proposition 1, we obtain

Theorem 5.

We leave the details to the reader.



54 Violeta Petkova CUBO
14, 3 (2012)

4 Spectral results for multipliers on Banach space of func-

tions on Zk

Let F(Zk) be the space of sequences of Zk with a finite number of not vanishing terms. Let E be
a Banach space of sequences on Zk satisfying the following conditions:

(h1) F(Z
k) is dense in E.

(h2) For every n ∈ Zk, the application E ∋ x −→ x(n) ∈ Ck is continuous.
(h3) For every z ∈ Tk, we have ψz(E) ⊂ E and supz∈Tk ‖ψz‖ < +∞, where

(ψz(x))(n1, ...,nk) = x(n1, ...,nk)z
n1
1 ...z

nk
k , ∀n ∈ Z

k, x ∈ E.

Denote by µ(E) the space of bounded operators on E commuting with the translations. Denote
by Si the operator of translation by ei, where ei(n) = 1, if ni = 1 and nj = 0, for j 6= i and else
ei(n) = 0. Suppose that the operator Si is bounded on E for all i ∈ Z. For M ∈ µ(E) define

M̃(z) =
∑

n∈Zk

M̂(n1, ...,nk)z
n1
1 ...z

nk
k ,

for z = (z1, ...,zk) ∈ Ck, where M̂(n1, ...,nk) = M(e0)(n1, ...,nk). For a ∈ E, set

ã(z) =
∑

n∈Zk

a(n)z
n1
1
...z
nk
k
, ∀z ∈ Ck.

Notice that for M ∈ µ(E) and a ∈ F(Zk), we have

Ma = M̂ ∗ a, ∀a ∈ F(Zk)

and formally we get

M̃a(z) = M̃(z)ã(z), a ∈ E, z ∈ Ck.

If φ ∈ F(Zk) denote by Mφ the operator given by Mφf = φ ∗ f, ∀f ∈ E. Define the set

ZkE =
{
z ∈ Ck,

∣∣∣
∑

n∈Zk

φ(n)z
n1
1
..z
nk
k

∣∣∣ ≤ ‖Mφ‖, ∀φ ∈ F(Zk)
}
.

Denote by σA(B1, ...,Bp) the joint spectrum of the elements B1,...,Bp in a commutative Banach

algebra A. Recall that σA(B1, ...,Bp) is the set of (λ1, ...,λp) ∈ Cp such that for all L ∈ A, the
operator (B1 −λ1I)L+...+(Bp −λpI)L is not invertible (see [13]). We have also the representation

σA(B1, ...,Bp) \ {0} = {(γ(B1), ...,γ(Bp)) : γ isacharacteronA}.

It is clear that

σA(B1, ...,Bp) ⊂ σ(B1) × ... × σ(Bp),

but in general these two sets are not equal and the inclusion could be strict.



CUBO
14, 3 (2012)

Spectral results for operators commuting with translations ... 55

Definition 4. Denote by A the closure of the subalgebra generated by the operators Mφ, φ ∈ F(Zk),
with respect to the operator norm topology.

Proposition 2. We have σA(S1, ....,Sk) \ {0} = ZkE \ {0}.

Proof. Let z ∈ Ck be such that z = (γ(S1), ...,γ(Sk)), where γ is a character on the algebra
A. Then, we have

∑

n∈Zk

φ(n)z
n1
1
...z
nk
k

=
∑

n∈Zk

φ(n)γ(S1)n1...γ(Sk)nk = γ(Mφ),∀φ ∈ F(Zk)

and it is clear that |γ(Mφ)| ≤ ‖Mφ‖, ∀φ ∈ F(Zk). It follows that σA(S1, ....,Sk) ⊂ ZkE. On the
other hand, if z ∈ ZkE \ {0}, we define

γz : Mφ −→
∑

n∈Zk

φ(n)z
n1
1
...z
nk
k
.

The application γz is a character on A and this implies that z = (γz(S1), ...,γz(Sk)) is in the joint
spectrum of S1, ...,Sk in A. So we have ZkE \ {0} = σA(S1, ....,Sk) \ {0}. ✷

Define

IE = {r ∈ Rk, r1T × ... × rkT ∈
◦

ZkE }.

For a ∈ E and r ∈ Ck, denote by (a)r the sequence

(a)r(n1, ...,nk) = a(n1, ...,nk)r
n1
1
...r
nk
k
, ∀(n1, ...,nk) ∈ Zk.

The following theorem was established in [7] (Theorem 4 and Collorary 1).

Theorem 6. Let E be a Banach space of sequences on Zk satisfying (h1), (h2) and (h3) and such
that Si is bounded on E for all i ∈ Z. Suppose that

◦

ZkE 6= ∅. Then, for M ∈ µ(E), there exists
θM ∈ H∞(

◦

ZkE) such that for f ∈ F(Zk) we have M̃f(z) = θM(z)f̃(z), ∀z ∈
◦

ZkE.

Following Lemma 2 in [7] there exists a sequence (Mm)m∈N ⊂ µ(E) such that:
limm→+∞ ‖Mma − Ma‖ = 0, ∀a ∈ E, Mm = Mφm, where φm ∈ F(Zk) and ‖Mm‖ ≤ C‖M‖.
Notice that using the sequence (Mm) and the same arguments as in the proof of Lemma 2, we

obtain that in fact θM = M̃ and, moreover, we get the following

Lemma 6. For M ∈ µ(E) and for f ∈ E such that (f)r ∈ l2(Zk), for all r ∈ IE we have

M̃f(z) = M̃(z)f̃(z), ∀z ∈
◦

ZkE.

Now we obtain the following spectral result.

Theorem 7. 1) For M ∈ µ(E), we have M̃(
◦

Zk
E
) ⊂ σ(M).

2) For M ∈ A, we have σ(M) \ {0} ⊂ M̃(ZkE) ⊂ σ(M).



56 Violeta Petkova CUBO
14, 3 (2012)

Proof. Let M ∈ µ(E). Suppose that α /∈ σ(M). Then we have K = (M − αI)−1 ∈ µ(E) and

K̃f(z) = K̃(z)f̃(z), ∀z ∈
◦

Z
k
E, ∀f ∈ E s.t. (f)r ∈ l2(Zk), ∀r ∈ IE. (14)

(
˜(M − αI)−1f

)
(z) =

( ∑

n∈Z

̂(M − αI)−1(n)zn
)( ∑

n∈Z

f(n)zn
)
, ∀f ∈ E,s.t. ∀r ∈ IE, (f)r ∈ l2(Zk).

If g ∈ F(Zk), following Lemma 6, we may replace f by (M − αI)g in (14). We get

g̃(z) =
( ∑

n∈Z

̂(M − αI)−1(n)zn
)( ∑

n∈Z

((M − αI)g)(n)zn
)

= K̃(z)(M̃g(z) − αg̃(z)) = K̃(z)(M̃(z) − α)g̃(z),

for all z ∈
◦

ZkE. This implies that for fixed r ∈ IE, K̃(rη)(M̃(rη) − α) = 1, ∀η ∈ Tk. Since, K̃ is
continuous on

◦

ZkE, we obtain that M̃(z) 6= α, for every z ∈
◦

ZkE. We conclude that

M̃(
◦

ZkE) ⊂ σ(M),

which proves part 1). Now suppose that M = Mφ, with φ ∈ F(Zk). Let λ ∈ σ(Mφ) \ {0}. Then
there exists γ a character on µ(E) such that

λ = γ(Mφ) =
∑

n∈Zk

φ(n)γ(Sn1,...,nk) = φ̃(γ(S1), ...,γ(Sk)) ∈ φ̃(ZkE).

The end of the proof of 2) is now very similar to the proof of 2) in Theorem 2 and is left to the

reader. ✷

Received: December 2011. Revised: January 2012.

References

[1] D.E. Edumnds and A. Nekvinda, Averaging operators on l{pn} and Lp(x), Math. Inequal.

Appl., 5, No. 2 (2002), p.235-246.

[2] F. Fernanda, Weighted shift operators and analytic function theory, Topics in Operator Theory

(C. Pearcy, ed.), Math. Surveys, No. 13, Amer. Math. Soc., Providence, RI, 1974, p.49-128.

[3] J. Lindenstrauss, L. Tzafriri, On Orlicz sequence spaces, Israel. J. Math. 10 (1971), p.379-390.

[4] A. Nekvinda, Equivalence of l{pn} norms and shift operators, Math. Inequal. Appl. 5, No. 4

(2002), p.711-723.

[5] V. Petkova, Wiener-Hopf operators on L2ω(R
+), Arch. Math.(Basel), 84 (2005), p.311-324.

[6] V. Petkova, Spectral theorem for multipliers on L2ω(R), Arch. Math. (Basel), 93 (2009), p.357-

368.



CUBO
14, 3 (2012)

Spectral results for operators commuting with translations ... 57

[7] V. Petkova, Multipliers on Banach spaces of functions on a locally compact abelian group, J.

London Math. Soc. 75 (2007), p.369-390.

[8] V. Petkova, Multipliers and Toeplitz operators on Banach spaces of sequences, Journal of

Operator Theory, 63 (2010), p.283-300.

[9] V. Petkova, Spectra of the translations and Wiener-Hopf operators on L2ω(R
+), Proc. Amer.

Math. Soc. to appear.

[10] V. Petkova, Multipliers and Wiener-Hopf operators on weighted Lp spaces, Central European

Journal of Mathematics, to appear., (arXiv:1112.4985v1).

[11] W. C. Ridge, Approximative point spectrum of a weighted shift, Trans. AMS, 147 (1970),

p.349-356.

[12] W. C. Ridge, Spectrum of a composition operator, Proc. AMS, 37 (1973), p.121-127.

[13] W. Zelasko, Banach Algebra, Elvesier Publishing Company, Amsterdam (1973).