Zeitschrift für Analysis und ihre Anwendungen

Journal for Analysis and its Applications Volume 19 (2000), No. 1, 3–12

Geometry of Numerical Ranges in Locally m-Convex ∗-Algebras

Th. Chryssakis

Abstract. In this paper we examine the symmetry of numerical ranges in a unital locally mconvex C^{∗}- algebra of a given element and its adjoint, with respect to a rotated real-axis, where the rotation angle depends on the value of the positive linear forms of the algebra (states) at the unit element of the algebra.

Keywords: Numerical, k-numerical, matrix numerical and Bonsal numerical ranges, k-states, locally m-convex, ∗- and C^{∗}-algebras, ∗-representations

AMS subject classification: 46 H 99, 46 K 99

0. Introduction

Given a unital locally m-convex ∗-algebra E the union of the (Bonsall) numerical ranges of an element a ∈ E and its adjoint a^{∗}, V (E, a)∪V (E, a^{∗}), contains a subset, symmetric with respect to the real axis ℝ. In the particular case that E is a locally m-convex C^{∗}- algebra (more specifically, locally C^{∗}-algebra in the terminology of [8]), the previous symmetric subset is the union itself, as before. As a consequence, one thus infers the symmetry with respect to ℝ of V (E, a), for any self-adjoint element a ∈ E, with E a locally m-convex C^{∗}-algebra (in this connection see, for instance, [7: Proposition 3.2]). In this symmetry of the numerical range, a crucial role is assigned to the “normalized states” of the algebra (i.e. continuous positive linear forms f of E, with f(1_{E}) = 1). Thus, by changing to generalized normalized states, that is, to similar forms, as before, where now f(1_{E}) = k ∈ ℂ, this implies the appearance of a rotation angle equal to arg k. So the previous symmetry properties of the numerical range are still in force, with respect to the new rotated axis. The justification of the latter symmetry properties constitute thus our main objective of this paper.

I want to express my sincere thanks to the referee, for his assiduous reading of the manuscript and the constructive remarks, that led to the present form of the paper.

Th. Chryssakis: Dept. Math. Univ., Panepistimiopolis, Athens 15784, Greece ISSN 0232-2064 / $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag Berlin

1. Preliminaries

Throughout this paper all algebras are complex and the topological spaces are always assumed to be Hausdorff.

A locally m-convex algebra (E, Γ ≡ {p_{α}}_{α∈I}) is a complex algebra E which is also a topological vector space, the topology of which is defined by an upper directed family of submultiplicative semi-norms Γ ≡ {p_{α}}_{α∈I}. If a locally m-convex algebra E has a continuous involution “*”, then E is said to be a locally m-convex ∗-algebra, and if the involution has the C^{∗}-property (i.e. p_{α}(x^{∗}x) = p_{α}(x)^{2} for all x ∈ E and α ∈ I), then we speak about a locally m-convex C^{∗}-algebra. At the end, a locally m-convex Q-algebra is a unital locally m-convex algebra E whose set of invertible elements is an open subset of E (cf. [10; p. 43/Definition 6.2]).

On the other hand, if E is a ∗-algebra (i.e. an involutive algebra), then by a ∗- representation of E, ϕ : E → L(H_{ϕ}) we shall always mean a ∗-morphism of E into the C^{∗}-algebra L(H_{ϕ}) of all bounded linear operators on a Hilbert space H_{ϕ}. If E is furthermore a locally m-convex ∗-algebra, the continuity of ϕ will be always considered with respect to the norm-operator topology of L(H_{ϕ}). Moreover, we denote by P(E) the set of all continuous positive linear forms on E, that is

P(E) = {f ∈ E′ : f(x^{∗}x) ≥ 0 for all x ∈ E}.

On the other hand, if E has a unit 1_{E} and for an f ∈ P(E) we have f(1_{E}) = 1, then we speak of a normalized continuous positive linear form on E, and if there is a vector ξ ∈ H_{ϕ} such that {ϕ(x)ξ : x ∈ E}^{−} = H_{ϕ} (the ”bar” here means ”topological closure”), then ϕ is called cyclic and the vector ξ is called a cyclic vector for ϕ.

Now, let (E, Γ ≡ {p_{α}}_{α∈I}) be a locally m-convex algebra with a unit 1_{E} and k ∈ ℂ such that 0 < |k| ≤ 1. The family of semi-norms Γ_{1} = \{p_{α}, \frac{1}{|k|}p_{α}\}_{α∈I} defines on E the same topology as that of the family Γ. In fact, if T and T_{1} are the topologies which are defined on E by the families Γ and Γ_{1}, respectively, then obviously T ⊆ T_{1} (T_{1} is finer than T). Conversely, let

V ≡ V_{α_{1},...,α_{n},ε_{1}} = {x ∈ E : q_{α_{i}}(x) ≤ ε_{1} (1 ≤ i ≤ n)}

be a neighborhood of 0 ∈ E for the topology T_{1} where q_{α_{i}} = p_{α_{i}} or q_{α_{i}} = \frac{1}{|k|}p_{α_{i}}. Then the neighborhood of 0 ∈ E for the topology T,

U ≡ U_{α_{1},...,α_{n},ε} = {x ∈ E : p_{α_{i}}(x) ≤ ε ≡ |k|ε_{1} (1 ≤ i ≤ n)}

is obviously contained to V, that is T_{1} ⊆ T and hence T = T_{1}. So without loss of generality we may assume that the family of semi-norms Γ ≡ {p_{α}}_{α∈I} which defines the locally m-convex topology on E, with every semi-norm p_{α} ∈ Γ contains the seminorm \frac{1}{|k|}p_{α} (α ∈ I), too, where k is an arbitrary but constant complex number with 0 < |k| ≤ 1.

2. The k-numerical range

Let (E, {p_{α}}_{α∈I}) be a unital locally m-convex algebra. Moreover, let E′ be the weak topological dual space of E and k ∈ ℂ with 0 < |k| ≤ 1. We consider the sets

(U_{α}(1))^{0} = {f ∈ E′ : |f(x)| ≤ 1 (x ∈ U_{α}(1))}  (2.1)

where U_{α}(1) = {x ∈ E : p_{α}(x) ≤ 1},

D_{k}^{α}(E, 1_{E}) = {f ∈ (U_{α}(1))^{0} : f(1_{E}) = k}  (2.2)

and

D_{k}(E, 1_{E}) = ⋃_{α∈I} D_{k}^{α}(E, 1_{E})  (2.3)

where 1_{E} is the unit of E. The elements of the set (2.3) are called k-states of E. For k = 1 we set D_{1}^{α}(E, 1_{E}) = D_{α}(E, 1_{E}) and D_{1}(E, 1_{E}) = D(E, 1_{E}).

We call k-numerical range of an element a ∈ E the set

V_{k}(E, a) = â(D_{k}(E, 1_{E})) = {f(a) : f ∈ D_{k}(E, 1_{E})}  (2.4)

where

â : D_{k}(E, 1_{E}) ⊆ E′ → ℂ, f → â(f) := f(a)

is the generalization of the Gel’fand transform of a. The numbers,

ν_{k}(E, a) ≡ ν_{k}(a) := sup{|λ| : λ ∈ V_{k}(E, a)}  (2.5)

is called the k-numerical radius of a. For k = 1 we set V_{1}(E, a) = V (E, a) and ν_{1}(E, a) = ν(E, a).

Now, if λ ∈ V (E, a) (a ∈ E), there is an f ∈ E′ with f(1_{E}) = 1 and |f(x)| ≤ p_{α}(x) (x ∈ E), for some α ∈ I, such that λ = f(a). So the linear form g = kf belongs to the set D_{k}^{α}(E, 1_{E}) ⊂ D_{k}(E, 1_{E}) and hence kλ = g(a) ∈ V_{k}(E, a), that is k · V (E, a) ⊂ V_{k}(E, a).

On the other hand, if ρ ∈ V_{k}(E, a), then ρ = g(a) for some g ∈ E′ with g(1_{E}) = k and |g(x)| ≤ p_{α}(x) \}(x ∈ E) for some α ∈ I. So |(\frac{1}{k} g) (x)| ≤ \frac{1}{|k|}p_{α}(x) = p_{β}(x) (x ∈ E) for some other β ∈ I (see the comments before Section 2) which means that the linear form f = \frac{1}{k} g ∈ E′ belongs to the set D_{1}^{β}(E, 1_{E}) ⊆ D(E, 1_{E}), hence f(a) ∈ V (E, a). So ρ = g(a) ∈ k · V (E, a), that is V_{k}(E, a) ⊆ k · V (E, a).

Now by the above we have

k · V (E, a) = V_{k}(E, a)  and  |k| · ν(E, a) = ν_{k}(E, a).  (2.6)

On the other hand, let (E_{α} = E/ker p_{α})_{α∈I} be the Arens-Michael decomposition of E (E = lim_{←} E_{α}, see [10: p. 88/Theorem 3.1]). Then, D_{k}^{α}(E, 1_{E}) is isomorphic to D_{k}(E_{α}, 1_{α}) where 1_{α} = 1_{E} + ker p_{α} is the unit of the normed algebra E_{α} = E/ker p_{α} (α ∈ I), so that D_{k}(E, 1_{E}) = ∪_{α∈I}D_{k}(E_{α}, 1_{α}) by (2.3) and hence

V_{k}(E, a) = ⋃_{α∈I} V_{k}(E_{α}, a_{α})  (2.7)

and

ν_{k}(E, a) = sup_{α∈I} ν_{k}(E_{α}, a_{α})  (2.8)

where a_{α} = a + ker p_{α} ∈ E_{α} (α ∈ I). In this respect we have the following

Proposition 2.1. Let (E, Γ ≡ {p_{α}}_{α∈I}) be a unital complete locally m-convex algebra. Then:

(i) For each α ∈ I, the set D_{k}^{α}(E, 1_{E}) ⊆ E′ is convex and compact.
(ii) For each a ∈ E, the set V_{k}(E, a) ⊆ ℂ is convex.

(iii) For each a ∈ E, the set V_{k}(E, a) ⊆ ℂ is bounded if and only if sup_{α∈I} p_{α}(a) < +∞.

Moreover, if the family {p_{α}}_{α∈I} is finite, then V_{k}(E, a) is a compact subset of ℂ.

Proof. (i) The set D_{k}^{α}(E, 1_{E}) ⊆ E′ is convex and closed since the sets (U_{α}(1))^{0} and ˆ1_{E}^{−1}({k}) are such. Moreover, the set D_{k}^{α}(E, 1_{E}) is equicontinuous since it is contained in the polar of a neighborhood of 0 ∈ E. So by the Alaoglu-Bourbaki theorem it is relatively compact. Hence D_{k}^{α}(E, 1_{E}) (α ∈ I) is a compact subset of E′.

(ii) For f, g ∈ D_{k}(E, 1_{E}) there are α, β ∈ I such that |f(x)| ≤ p_{α}(x) and |g(x) ≤ p_{β}(x) (x ∈ E). Since the family of semi-norms Γ ≡ {p_{α}}_{α∈I} is upper directed, there is γ ∈ I such that f, g ∈ D_{k}^{γ}(E, 1_{E}) ⊆ (U_{γ}(1))^{0}. Now, if h is a convex combination of f and g, then by the convexity of D_{k}^{γ}(E, 1_{E}) we have h ∈ D_{k}^{γ}(E, 1_{E}) ⊆ D_{k}(E, 1_{E}), that is D_{k}(E, 1_{E}) ⊆ E′ is a convex subset, and by (2.4) V_{k}(E, a) ⊆ ℂ is also a convex subset.

(iii) Let (E_{α} ≡ E/ker p_{α})_{α∈I} be the Arens-Michael decomposition of E. Then for the normed algebras E_{α} (α ∈ I) we have

ν(E_{α}, a_{α}) ≥ \frac{1}{e}||a_{α}||_{α} = \frac{1}{e}p_{α}(a)  (2.9)

(cf. [1: p. 34/Theorem 1]) and

ν_{k}(E_{α}, a_{α}) ≤ ||a_{α}||_{α} = p_{α}(a).  (2.10)

By (2.9), (2.6) and (2.10) we get

\frac{|k|}{e}p_{α}(a) ≤ ν_{k}(E_{α}, a_{α}) ≤ p_{α}(a)

and taking the suprema we get

\frac{|k|}{e} sup_{α} p_{α}(a) ≤ sup_{α} ν_{k}(E_{α}, a_{α}) ≤ sup_{α} p_{α}(a)

so that, by (2.8),

\frac{|k|}{e} sup_{α} p_{α}(a) ≤ ν_{k}(E, a) ≤ sup_{α} p_{α}(a)

which means that V_{k}(E, a) ⊆ ℂ is bounded if and only if sup p_{α}(a) < +∞. On the other hand, if the family of semi-norms Γ ≡ {p_{α}}_{a∈I} is finite, then the semi-norm q(x) = max_{α∈I} p_{α}(x) (x ∈ E) defines on E the same topology as the family Γ, and D_{k}(E, 1_{E}) = D_{k}^{q}(E, 1_{E}). Since D_{k}^{q}(E, 1_{E}) is convex and compact one has that V_{k}(E, a) (a ∈ E) has the same property

Now, let E be a unital locally m-convex algebra and F be a subalgebra of E. Then, for every a ∈ E,

V_{k}(E, a) = V_{k}(F, a).  (2.11)

In fact, since the map f ↦ f|_{F} : D_{k}(E, 1_{E}) → D_{k}(F, 1_{E}) is “onto”, by the Hahn-Banach theorem, (2.4) implies (2.11).

Scholium. From the above it follows that the numerical range of an element a ∈ E does not change replacing E by its completion Ê. Thus, without loss of generality we may assume that the initial locally m-convex algebra E is complete. Moreover, V_{k}(E, a) may be computed from the subalgebra F generated by a and 1_{E}, or from the closure of F.

Proposition 2.2. Let (E, Γ ≡ {p_{α}}_{α∈I}) be a unital locally m-convex algebra. Then the set H_{k}(E) of strongly k-Hermitian elements (i.e. elements with real k-numerical range) is closed.

Proof. Let (x_{δ})_{δ∈J} ⊆ H_{k}(E) be a net of strongly k-Hermitian elements of E with x_{δ} ⟶_{δ} x. For λ ∈ V_{k}(E, x) there is an α ∈ I and an f ∈ (U_{α}(1))^{0} with f(1_{E}) = k such that λ = f(x). On the other hand, the numbers λ_{δ} = f(x_{δ}) (δ ∈ J) are reals and

|λ_{δ} − λ| = |f(x_{δ} − x)| ≤ p_{α}(x_{δ} − x) for f ∈ (U_{α}(1))^{0}.

Since p_{α}(x_{δ} − x) ⟶_{δ} 0 we have λ ∈ ℝ, that is x ∈ H_{k}(E)

Theorem 2.3. Let (E, Γ ≡ {p_{α}}_{α∈I}) be a unital locally m-convex algebra and J a closed two-sided ideal of E. For the locally m-convex quotient algebra (E/J, Γ̇ ≡ {ṗ_{α}}_{α∈I}) we have

V_{k}(E/J, π(a)) = ⋂_{b∈J} V_{k}(E, a + b)  (a ∈ E)  (2.12)

where π : E → E/J is the canonical quotient map.

For the proof of Theorem 2.3 we need the following

Lemma 2.4. Let (E, Γ ≡ {p_{α}}_{α∈I}) be a unital locally m-convex algebra and a ∈ E. Then

V_{k}(E, a) = ⋃_{α∈I} ( ⋂_{z∈ℂ} {λ ∈ ℂ : |λ − kz| ≤ p_{α}(a − z 1_{E})}).  (2.13)

Proof. Let λ ∈ ℂ satisfy the inequality

|λ − kz| ≤ p_{α}(a − z 1_{E})  (z ∈ ℂ)  (2.14)

for some α ∈ I. If a = z_{0}1_{E} for some z_{0} ∈ ℂ, then from (2.14) we get λ = kz_{0}. So for every f ∈ D_{k}^{α}(E, 1_{E}) we have λ = z_{0}f(1_{E}) = f(a) ∈ V_{k}(E, a).

On the other hand, if the elements a and 1_{E} are linearly independent, we define the function

f_{0} : M ≡ L(a, 1_{E}) → ℂ, μa + ν1_{E} ↦ f_{0}(μa + ν1_{E}) := μλ + νk

where M is the subspace of E generated by a and 1_{E}. Then

\begin{matrix}|f_{0}(μa + ν1_{E})| & = |μλ + νk| \\ & = |μ| |λ − k( − \frac{ν}{μ})| \\ & ≤ |μ| p_{α}(a − ( − \frac{ν}{μ})1_{E}) \\ & = p_{α}(μa + ν1_{E}) \end{matrix} (μ, ν ∈ ℂ, μ ≠ 0).

Now, the Hahn-Banach theorem guarantees that there is an f ∈ E′ with |f(x)| ≤ p_{α}(x) (x ∈ E) and f|_{M} = f_{0}, so that f ∈ D_{k}^{α}(E, 1_{E}) and λ = f_{0}(a) = f(a) ∈ V_{k}(E, a).

Conversely, let λ ∈ V_{k}(E, a). Then there are α ∈ I and f ∈ (U_{α}(1))^{0} with f(1_{E}) = k such that λ = f(a). So we have |λ − kz| = |f(a − z1_{E})| ≤ p_{α}(a − z1_{E}) for every z ∈ ℂ

Proof of Theorem 2.3. Let λ, z ∈ ℂ be such that

|λ − kz| ≤ ṗ_{α}(π(a) − z ˙1_{E})  (2.15)

for some α ∈ I where ˙1_{E} = π(1_{E}) = 1_{E} + J is the unit of the quotient algebra E/J. Since the canonical map π is a homomorphism we have

ṗ_{α}(π(a) − z1_{E}) = ṗ_{α}(π(a − z1_{E})) := inf_{b∈I} p_{α}((a + b) − z1_{E})

so that by (2.15) |λ − kz| ≤ p_{α}((a + b) − z1_{E}) (b ∈ I) for some α ∈ I. Hence, Lemma 2.4 implies (2.12)

3. Geometry of the k-numerical range

In this section we prove some elegant geometrical properties of the k-numerical range (see Introduction). We first have the following

Theorem 3.1. Let (E, Γ ≡ {p_{α}}_{α∈I}) be a locally m-convex ∗-algebra with unit 1_{E}, a ∈ E, and k ∈ ℂ with |k| ≤ 1. Then the set V_{k}(E, a) ∪ V_{k}(E, a^{∗}) contains a subset A ⊂ ℂ symmetric with respect to the line ε = {re^{iθ_{0}} : r ∈ ℝ}, where θ_{0} ∈ [0, 2π] with cos θ_{0} = Re k and sin θ_{0} = Im k.

Proof. Let f ∈ D_{1}(E, 1_{E}) ≡ D(E, 1_{E}) be a normalized state of E which is further a positive linear form (i.e. f ∈ P(E)). Then there is a continuous cyclic ∗- representation of E, ϕ_{f} : E → L(H_{ϕ}), and a unital cyclic vector ξ ∈ H_{ϕ_{f}} such that f(x) = (ϕ_{f}(x)ξ|ξ) (x ∈ E) (cf. [3, 8]). We have

f(a^{∗}) = (ϕ_{f}(a^{∗})ξ|ξ) = (ϕ_{f}(a)^{∗}ξ|ξ) = (ξ|ϕ_{f}(a)ξ) = (\overline{ϕ_{f}(a)ξ|ξ}) = \overline{f(a)}.

So the set

B = {{f(a)} ∪ {f(a^{∗})} : f ∈ D(E, 1_{E}) ∩ P(E)} ⊆ ℂ

is symmetric with respect to the ℝ-axis, hence the set

A = e^{iθ_{0}}B ⊆ e^{iθ_{0}}(V (E, a) ∪ V (E, a^{∗})) = V_{k}(E, a) ∪ V_{k}(E, a^{∗})

(see (2.6)) is symmetric with respect to the line ε

Corollary 3.2. Let (E, Γ ≡ {p_{α}}_{α∈I}) be a unital locally m-convex ∗-algebra, a ∈ E self-adjoint, and k ∈ ℂ with |k| ≤ 1. Then the k-numerical range of a contains a subset of ℂ, symmetric with respect to the line ε = {re^{iθ_{0}} : r ∈ ℝ}, where θ_{0} ∈ [0, 2π] with cos θ_{0} = Re k and sin θ_{0} = Im k.

Corollary 3.3. Let (E, Γ ≡ {p_{α}}_{α∈I}) be a locally m-convex C^{∗}-algebra with unit 1_{E}, a ∈ E, and k ∈ ℂ with |k| ≤ 1. Then the set V_{k}(E, a) ∪ V_{k}(E, a^{∗}) ⊆ ℂ is symmetric with respect to the line ε = {re^{iθ_{0}} : r ∈ ℝ}, where θ_{0} ∈ [0, 2π] with cos θ_{0} = Re k and sin θ_{0} = Im k.

Proof. In every unital locally m-convex C^{∗}-algebra E we have D(E, 1_{E}) = P(E) (cf. [4: Theorem 3.7_{(i)⇔(ii)}]). So by the above Theorem 3.1 we have what we wanted to prove

By the above Corollary 3.3 we see that the value of the states of a unital locally m-convex C^{∗}-algebra E at the unit element 1_{E} determines the argument of the line, with respect to which the union of the numerical ranges of the elements a and a^{∗} is symmetric.

Proposition 3.4. Let (E, Γ ≡ {p_{α}}_{α∈I}) be a locally m-convex C^{∗}-algebra with unit 1_{E}, a ∈ E self-adjoint, and k ∈ ℂ with |k| ≤ 1. Then the k-numerical range of a is a convex subset of the line ε = {re^{iθ_{0}} : r ∈ ℝ}, where θ_{0} ∈ [0, 2π] with cos θ_{0} = Re k and sin θ_{0} = Im k.

Proof. Since the locally m-convex ∗-algebra E has the C^{∗}-property, we have D(E, 1_{E}) = P(E) (cf. [4: Theorem 1.7_{(i)⇔(ii)}]). Now, having in mind the proof of Theorem 3.1, we have f(a) ∈ ℝ for every f ∈ D(E, 1_{E}) = P(E). So V (E, a) ⊂ ℝ, hence V_{k}(E, a) = e^{iθ_{0}}V (E, a) is a convex subset of the line ε

Corollary 3.5. Let (E, Γ ≡ {p_{α}}_{α∈I}) be a unital locally m-convex Q-∗-algebra with the C^{∗}-property, a ∈ E self-adjoint, and k ∈ ℂ with |k| ≤ 1. Then the k-numerical range of a is a line segment of the line ε = {re^{iθ_{0}} : r ∈ ℝ} where θ_{0} ∈ [0, 2π] with cos θ_{0} = Re k and sin θ_{0} = Im k.

Proof. The proof is a consequence of Proposition 3.4, Proposition 2.1 and [8: Theorem 7.6_{(1)⇔(3)}]

Now, let M_{n}(ℂ) be the ∗-algebra of all n × n complex matrices with an involution defined by A^{∗} = \overline{^{t}A} (A ∈ M_{n}(ℂ)), where t means transpose and overline means complex conjugation. If L(ℂ^{n}) is the C^{∗}-algebra of all (bounded) operators on the numerical space ℂ^{n}, with inner product (a|b) := \overline{^{t}b}a ≡ b^{∗}a (a, b ∈ ℂ^{n}), then by considering each n × n matrix A ∈ M_{n}(ℂ) ≅ L(ℂ^{n}) as an operator on ℂ^{n} (i.e. A : ℂ^{n} → ℂ^{n}, x ↦ A(x) := Ax, x = (x_{1}, …, x_{n})^{t} ∈ ℂ^{n}) and taking k = 1, the classical numerical range of the matrix A is a (convex) subset of the k-numerical range V_{k}(M_{n}(ℂ), A) ≡ V_{1}(M_{n}(ℂ), A) ≡ V (M_{n}(ℂ), A) of the element A ∈ M_{n}(ℂ) ≅ L(ℂ^{n}) (cf. [7: Comments after Theorem 2.1] and [6: Theorem 2.1]). So by the above Corollary 3.5 we have that the classical numerical range of a Hermitian matrix A is a line segment of the line ℝ.

On the other hand, if A ∈ M_{n}(ℂ) is an n × n matrix such that the matrix e^{−iθ_{0}}A (θ_{0} ∈ [0, 2π]) is Hermitian, then by Corollary 3.5 the classical numerical range W (A) of A is a line segment of the line ε = {r · e^{iθ_{0}} : r ∈ ℝ}, since W (A) is a bounded convex subset of ℂ and

W (A) = e^{iθ_{0}}W (e^{−iθ_{0}}A) ⊆ e^{iθ_{0}}V (M_{n}(ℂ), e^{−iθ_{0}}A) = V_{k}(M_{n}(ℂ), e^{−iθ_{0}}A) ⊆ ε

where k ∈ ℂ is such that Re k = cos θ_{0} and Im k = sin θ_{0}.

Now, if A ∈ M_{n}(ℂ) is an n×n matrix with elements on the line ε = {r·e^{iθ_{0}} : r ∈ ℝ}, θ_{0} ∈ [0, 2π], then by Corollary 3.3 and the relation W (A) = W (^{t}A) (see [12: p. 7]) W (e^{−iθ_{0}}A) is symmetric with respect to the R-axis, hence W (A) = e^{iθ_{0}}W (e^{−iθ_{0}}A) is symmetric with respect to the line ε. That is we take [11: Proposition 2.1 and its Corollary].

4. The k-spatial numerical range

Let (E, Γ ≡ {p_{α}}_{α∈I}) be a locally convex space, T ∈ L(E) a continuous operator on E, and k ∈ ℂ with |k| ≤ 1. For p_{α} ∈ Γ and x ∈ U_{α}(1) ≡ {x ∈ E : p_{α}(x) ≤ 1} we consider the sets

D_{k}^{α}(E, x) = {f ∈ (U_{α}(1))^{0} : f(x) = k}  (4.1)
V_{k}^{α}(T, x) = {f(T x) : f ∈ D_{k}^{α}(E, x)}  (4.2)
V_{k}^{α}(T) = ⋃_{x∈S_{α}(1)} V_{k}^{α}(T, x).  (4.3)

where S_{α}(1) = {x ∈ E : p_{α}(x) = 1}. Applying an analogous argument as in the proof of Proposition 2.1, we can prove that for each x ∈ U_{α}(1) the set D_{k}^{α}(E, x) ⊆ E′_{s} is (weakly) compact and convex.

Definition 4.1. Let (E, Γ ≡ {p_{α}}_{α∈I}) be a locally convex space and T ∈ L(E). We call k-spatial numerical range of T the set

V_{k}(T) = ⋃_{α∈I} V_{k}^{α}(T).  (4.4)

In this regard, we have the following

Lemma 4.1. Let (E, Γ ≡ {p_{α}}_{α∈I}) be a locally convex space, T ∈ L(E) and x ∈ E with p_{α}(x) = 1} for some α ∈ I. Then

V_{k}^{α}(T, x) = ⋂_{ζ∈ℂ} {λ ∈ ℂ : |λ − kζ| ≤ p_{α}((T − ζI_{E})x)}  (4.5)

where k ∈ ℂ with |k| ≤ 1.

Proof. We apply an analogous argument as in the proof of Lemma 2.4

Theorem 4.2. Let (E, Γ ≡ {p_{α}}_{α∈I}) be a locally m-convex algebra with unit 1_{E}, such that p_{α}(1_{E}) = 1 (α ∈ I). Moreover, for some a ∈ E consider the operator

T_{a} : E → E,  x ↦ T_{a}(x) := ax.  (4.6)

Then

V_{k}(E, a) = V_{k}(T_{a}).  (4.7)

Proof. The operator T_{a} is clearly continuous since p_{α}(T_{a}x) = p_{α}(ax) ≤ p_{α}(a)p_{α}(x) (x ∈ E). Now, for λ ∈ V_{k}(T_{a}) there is an α ∈ I such that |λ − kζ| ≤ p_{α}((T_{a} − ζI)x) for all ζ ∈ ℂ. But

p_{α}((T_{a} − ζI)x) = p_{α}((a − ζ1_{E})x) ≤ p_{α}(a − ζ1_{E})p_{α}(x) = p_{α}(a − ζ1_{E})  (4.8)

for all ζ ∈ ℂ, so that |λ − kζ| ≤ p_{α}(a − ζ1_{E}) for all ζ ∈ ℂ. Thus Lemma 2.4 implies that λ ∈ V_{k}(E, a).

On the other hand, if λ ∈ V_{k}(E, a), relation (4.8) is true for some α ∈ I and for all ζ ∈ ℂ. So the equality p_{α}(a − ζ1_{E}) = p_{α}((T_{a} − ζI)1_{E}) and Lemma 4.1 imply that λ ∈ V_{k}^{α}(T_{a}, 1_{E}), hence λ ∈ V_{k}^{α}(T_{a}) ⊆ V_{k}(T_{a})

For convenience, we recall the definition of an upper semi-continuous map.

Definition 4.2. Let E and F be two topological spaces and let 2^{F} be the set of all subsets of F. A map ϕ : E → 2^{F} is said to be upper semi-continuous if for each x ∈ E and each neighborhood U of ϕ(x), there exists a neighborhood V of x such that y ∈ V implies ϕ(y) ⊆ U.

In this regard, we now have the following

Lemma 4.3 (see [5: Lemma 3.6]). Let E and F be topological spaces with F compact. Let ϕ : E → 2^{F} be a map such that for every x ∈ E the set ϕ(x) ⊆ F is closed. Then ϕ is upper semi-continuous if and only if for any nets (x_{δ}) ⊆ E and (y_{δ}) ⊆ ϕ(x_{δ}) with lim_{δ} x_{δ} = x and lim_{δ} y_{δ} = y one has y ∈ ϕ(x).

Theorem 4.4. Let (E, Γ ≡ {p_{α}}_{α∈I}) be a locally convex space and suppose that for some α ∈ I the set S_{α}(1) = {x ∈ E : p_{α}(x) = 1} is bounded. If T is a continuous linear operator on E, then the map

x ↦ V_{k}^{α}(T, x),  S_{α}(1) → 2^{ℂ}  (4.9)

is upper semi-continuous and sends S_{α}(1) (α ∈ I) into non-void compact convex subsets of ℂ.

Proof. Let x ∈ S_{α}(1) and λ ∈ V_{k}^{α}(T, x). There is an f ∈ D_{k}^{α}(E, x) ⊆ (U_{α}(1))^{0} such that λ = f(T x), so that |λ| ≤ p_{α}(T x) (x ∈ E) and by the continuity of T there are β ∈ I and μ > 0 such that p_{α}(T x) ≤ μp_{β}(x) (x ∈ E).

On the other hand, since S_{α}(1) is bounded there is a ξ > 0 such that S_{α}(1) ⊆ ξV, where V = {x ∈ E : p_{β}(x) ≤ 1} (cf. [9: p. 109]). So for x ∈ S_{α}(1) we have p_{β}(x) ≤ ξ, hence |λ| ≤ μξ ≡ ρ, that is V_{k}^{α}(T, x) ⊆ {λ ∈ ℂ : |λ| ≤ ρ} ≡ S_{ρ}(ℂ) for every x ∈ S_{α}(1). Moreover, the sets V_{k}^{α}(T, x) (x ∈ S_{α}(1)) are compact and convex since the sets D_{k}^{α}(E, x) (x ∈ S_{α}(1)) are such (see the comments before Definition 4.1). Now we complete the proof by applying Lemma 4.3: Consider nets (x_{δ})_{δ∈J} ⊆ S_{α}(1) and (λ_{δ}) ⊆ ℂ with λ_{δ} ∈ V_{k}^{α}(T, x) ⊆ S_{ρ}(ℂ). If lim_{δ} x_{δ} = x and lim_{δ} λ_{δ} = λ, we have to prove that λ ∈ V_{k}^{α}(E, x). In fact, there is a net (f_{δ}) ⊆ E′ with f_{δ} ∈ D_{k}^{α}(E, x_{δ}) ⊆ (U_{α}(1))^{0} such that λ_{δ} = f_{δ}(T x) (δ ∈ J). On the other hand, by compactness of (U_{α}(1))^{0} ⊆ E′_{S}, there is a subnet (f_{n_{δ}})_{δ∈J} with lim_{δ} f_{n_{δ}} = f ∈ (U_{α}(1))^{0}. Thus, one obtains

|k − f(x)| ≤ |k − f_{n_{δ}}(x)| + |f_{n_{δ}}(x) − f(x)|
= |f_{n_{δ}}(x_{n_{δ}} − x)| + |(f_{n_{δ}} − f)(x)|
⟶_{δ} 0

which yields f(x) = k. Finally, we have

|λ − f(T x)| ≤ |λ − λ_{n_{δ}}| + |λ_{n_{δ}} − f_{n_{δ}}(T x)| + |f_{n_{δ}}(T x) − f(T x)|
= |λ − λ_{n_{δ}}| + |f_{n_{δ}}(T x_{n_{δ}} − T x)| + |(f_{n_{δ}} − f)(T x)|
⟶_{δ} 0

which gives λ = f(T x) ∈ V_{k}^{α}(T, x). So, by Lemma 4.3, the map (4.9) is upper semicontinuous

References

[1] Bonsall, F. F. and J. Duncan: Numerical Ranges of Operators on Normed Spaces and of Normed Algebras (London Math. Soc. Lect. Note Ser.: Vol. 2). Cambridge: Univ. Press 1971.

[2] Bonsall, F. F. and J. Duncan: Numerical Ranges, Vol. II (London Math. Soc. Lect. Note Ser.: Vol. 10). Cambridge: Univ. Press 1973.

[3] Brooks, R. M.: On locally m-convex ∗-algebras. Pacific J. Math. 23 (1967), 5 – 23.

[4] Chryssakis, Th.: Numerical ranges in locally m-convex algebras, Part I. J. Austral. Math. Soc. (Series A) 41 (1986), 304 – 316.

[5] Chryssakis, Th.: Spectra and numerical ranges in locally m-convex algebras. Bul. Greek Math. Soc. 35 (1993), 55 – 72.

[6] Chryssakis, Th.: ϕ-spatial numerical range in locally m-convex ∗-algebras. Math. Japonica 42 (1995), 511 – 516.

[7] Chryssakis, Th.: Symmetry of ϕ-spatial numerical ranges of algebra polynomials and of matrix polynomials. J. Math. Sci. 98/99 (invited paper).

[8] Fragoulopoulou, M.: An Introduction to the Representation Theory of Topological ∗- Algebras. Münster: Schriftenreihe Math. Inst. 48 (1988), 1 – 81.

[9] Horvath, J.: Topological Vector Spaces and Distributions. Reading (Mass.): Addison-Wesley 1966.

[10] Mallios, A.: Topological Algebras. Selected Topics. Amsterdam: North-Holland 1986.

[11] Maroulas, J. and P. Psarrakos: Geometrical properties of numerical range of matrix polynomials. Math. Appl. 31 (1996), 41 – 47.

[12] Psarrakos, P.: Numerical Range of Matrix Polynomials and Applications. Ph.D. Thesis. Athens: Nat. Techn. Univ. 1997.

Received 16.06.1998; in revised form 05.10.1999