Tensor Sum Operator and Dynamical System in Measurable Function Spaces

G. S. Vigneswari; P. Chandra Kala; P. Sujin

1. Introduction

Let X be a Hausdorff topological space, M(X, E) be the space of all measurable functions from X into E and C(X, E) be the vector subspace of M(X, E) consisting of the continuous functions f from X into E. Let V be a set of non-negative upper semicontinuous functions on X. If V is a set of weights on X such that, given any x ∈ X, there is some v ∈ V for which v(x) > 0, we write V > 0.

A set V of weights on X is said to be directed upward provided that, for every pair u₁, u₂ in V and α > 0, there exists v ∈ V such that αu_i ≤ v (pointwise on X) for i = 1, 2.

By a system of weights, we mean a set V of weights on X which additionally satisfies V > 0. Let cs(E) be the set of all continuous functions from X into E.

If V is a system of weights on X, then the pair (X, V) is called the weighted topological system. Associated with each weighted topological system (X, V), we have the weighted spaces of continuous E-valued functions defined as:

M V_{0} (X, E) = {f ⊞ g \in M (X, E) ⊞ M (X, E) : v q (f ⊞ g) vanishes at g on X for each v \in V, q \in cs (E)}

(1)

M V_{p} (X, E) = {f ⊞ g \in M (X, E) ⊞ M (X, E) : v q (f ⊞ g) in L^{p} for all v \in V, q \in cs (E)}

(2)

M V_{b} (X, E) = {f ⊞ g \in M (X, E) ⊞ M (X, E) : v q ((f ⊞ g) (x)) is bounded in E for all v \in V, q \in cs (E)}

(3)

Let v ∈ V, q ∈ cs(E) and f ⊞ g ∈ M(X, E) ⊞ M(X, E). If we define ‖f ⊞ g‖_v,q = sup {(∫_X(v(x)q(f ⊞ g)(x))^pdμ)^1/p for all x ∈ X}, then ‖.‖_v can be regarded as a seminorm on either MV₀(X, E) ⊞ MV₀(X, E) or MV_b(X, E) ⊞ MV_b(X, E), and the family {‖.‖_v,q: v ∈ V, q ∈ cs(E)} of seminorms defines a Hausdorff locally convex topology on each of these spaces. This topology will be denoted by w_v, and the vector spaces MV₀(X, E) and MV_b(X, E) endowed with w_v are called the weighted locally convex spaces of vector-valued continuous functions. It has a basis of closed absolutely convex neighbourhoods of the origin of the form

B_{v, q} = {f ⊞ g \in M V_{b} (X, E) ⊞ M V_{b} (X, E) : ∥ f ⊞ g ∥_{v, q} \leq 1}

(4)

Also, MV₀(X, E) is a closed subspace of MV_b(X, E).

1.1. Functions inducing tensor sum operators on weighted spaces of measurable functions

In this section, let us investigate the functions inducing tensor sum operators on weighted spaces of measurable functions.

Theorem 1.1.1. Let φ : X → X and π_t : X → ℂ be measurable functions. Then (C_φ ⊞ M_{π_t})(f ⊞ g) is a tensor sum operator for every t ∈ ℝ, f ⊞ g ∈ MV₀(X) ⊞ MV₀(X) iff V|C_φ ⊞ M_{π_t}| ≤ V.

Proof. First suppose V|C_φ ⊞ M_{π_t}| ≤ V. Then for all v ∈ V, there exists u ∈ V such that v|C_φ ⊞ M_{π_t}| ≤ u (pointwise on X). We show that C_φ ⊞ M_{π_t} is a continuous linear operator on MV₀(X) ⊞ MV₀(X). Clearly, C_φ ⊞ M_{π_t} is linear on MV₀(X) ⊞ MV₀(X). In order to prove the continuity of C_φ ⊞ M_{π_t} on MV₀(X) ⊞ MV₀(X), it is enough to show that C_φ ⊞ M_{π_t} is continuous at the origin. For this, suppose f_α ⊞ g_α is a net in MV₀(X) ⊞ MV₀(X) such that P_v(f_α ⊞ g_α) → 0 for every v ∈ V.

Now,

P_{v} (C_{φ} f_{α} ⊞ M_{π_{t}} g_{α}) = P_{v} (C_{φ} f_{α}) ⊞ P_{v} (M_{π_{t}} g_{α}) for all t \in R . = P_{v} (f_{α} \circ φ (x)) ⊞ P_{v} (π_{t} (x) g_{α} (x)) = (\int_{X} (v (x) q (f_{α} (φ (x))))^{p} d μ)^{\frac{1}{p}} ⊞ (\int_{X} (v (x) q (e^{t h (x)} g_{α} (x)))^{p} d μ)^{\frac{1}{p}} = (\int_{X} (v (x) q (f_{α} (x)))^{p} d μ)^{\frac{1}{p}} ⊞ (\int_{X} (v (x) q (e^{t h (x)} g_{α} (x)))^{p} d μ)^{\frac{1}{p}} = (\int_{X} (u (x) q (f_{α} (x)))^{p} d μ)^{\frac{1}{p}} ⊞ (\int_{X} (u (x) q (g_{α} (x)))^{p} d μ)^{\frac{1}{p}} as t \to 0 = P_{u} (f_{α} ⊞ g_{α}) \to 0

(5)

This proves the continuity of C_φ ⊞ M_{π_t} at the origin and hence C_φ ⊞ M_{π_t} is continuous on MV₀(X) ⊞ MV₀(X).

Conversely, suppose C_φ ⊞ M_{π_t} is a continuous linear operator on MV₀(X) ⊞ MV₀(X). We shall show that V|C_φ ⊞ M_{π_t}| ≤ V. Let v ∈ V. Since C_φ ⊞ M_{π_t} is continuous at the origin, there exists u ∈ V such that (C_φ ⊞ M_{π_t})(B_u) ⊆ B_v. We claim that v|C_φ ⊞ M_{π_t}| ≤ 2u. Take x₀ ∈ X and set u(x₀) = ε. In case ε > 0, N = {x ∈ X: u(x) < 2ε} is an open neighbourhood of x₀. Then there exists f ⊞ g ∈ MV₀(X) ⊞ MV₀(X) such that 0 ≤ f ⊞ g ≤ 1 and f ⊞ g(X − N) = 0.

Let h = (2ε)⁻¹(f ⊞ g). Then clearly h ∈ B_u. Since (C_φ ⊞ M_{π_t})(B_u) ⊆ B_v, we have (C_φ ⊞ M_{π_t})(h) ∈ B_v and this yields that v(x)|(C_φ ⊞ M_{π_t})(x)||h(x)| ≤ 1 for all x ∈ X. From this it follows that v(x)|(C_φ ⊞ M_{π_t})(x)||f ⊞ g(x)| ≤ 2ε for all x ∈ X.

Now suppose u(x₀) = 0 and that v(x₀)|(C_φ ⊞ M_{π_t})(x₀)| > 0. If we put ε = v(x₀)|(C_φ ⊞ M_{π_t})(x₀)|, which is not greater than two, and set N = {x ∈ X: u(x) < ε}, then N would be an open neighbourhood of x₀ and we could again find f ⊞ g ∈ MV₀(X) ⊞ MV₀(X) such that 0 ≤ f ⊞ g ≤ 1, f ⊞ g(x₀) = 1 and f ⊞ g(X − N) = 0. Now let h = ε⁻¹(f ⊞ g). Then clearly h ∈ B_u and (C_φ ⊞ M_{π_t})(h) ∈ B_v. Hence v(x)|(C_φ ⊞ M_{π_t})(x)||h(x)| ≤ 1 for all x ∈ X. This implies that v(x)|(C_φ ⊞ M_{π_t})(x)||f ⊞ g(x)| ≤ ε for all x ∈ X. From this it follows that v(x₀)|(C_φ ⊞ M_{π_t})(x₀)| ≤ v(x₀)|(C_φ ⊞ M_{π_t})(x₀)|/2, which is impossible. This proves our claim and hence the proof is complete.

Now we shall characterise tensor sum operators on MV₀(X, E) ⊞ MV₀(X, E) induced by scalar-valued and vector-valued functions.

1.2. Characterisation of tensor sum operators

In this section, let us investigate the characterisation of tensor sum operators.

Theorem 1.2.1. Let φ : X → X and π_t : X → ℂ be measurable functions. Then (C_φ ⊞ M_{π_t})(f ⊞ g) is a tensor sum operator for every t ∈ ℝ, f ⊞ g ∈ MV₀(X, E) ⊞ MV₀(X, E) iff V‖C_φ ⊞ M_{π_t}‖ ≤ V.

Proof. Similar to the proof of Theorem 1.1.1.

Theorem 1.2.2. Let E be a locally multiplicatively convex (lmc) algebra with unit e, and let φ : X → E and π_t : X → ℂ be bounded measurable functions. Then (C_φ ⊞ M_{π_t})(f ⊞ g) is a tensor sum operator for t ∈ ℝ, f ⊞ g ∈ MV₀(X, E) ⊞ MV₀(X, E) iff V_p ∘ (φ ⊞ π_t) ≤ V for all p ∈ P.

Proof. Suppose V_p ∘ (φ ⊞ π_t) ≤ V for all p ∈ P. Then for all v ∈ V, there exists u ∈ V such that v_p ∘ (φ ⊞ π_t) ≤ u (pointwise on X). We shall prove that the mappings φ : X → E and π_t : X → ℂ give rise to a linear transformation C_φ ⊞ M_{π_t} from MV₀(X, E) ⊞ MV₀(X, E) into itself, defined as C_φf = φf and M_{π_t}g = π_tg for every f ⊞ g ∈ MV₀(X, E) ⊞ MV₀(X, E), where the product is pointwise. To show that it is a continuous linear operator on MV₀(X, E), we shall establish the continuity of C_φ ⊞ M_{π_t} at the origin. For this, let {f_α ⊞ g_α} be a net in MV₀(X, E) ⊞ MV₀(X, E) such that, for all v ∈ V, p ∈ P, P_v,q(f_α ⊞ g_α) → 0.

Then

P_{v, q} (C_{φ} f_{α} ⊞ M_{π_{t}} g_{α}) = P_{v, q} (C_{φ} f_{α}) ⊞ P_{v, q} (M_{π_{t}} g_{α}) for all t \in R . = P_{v, q} (f_{α} \circ φ (x)) ⊞ P_{v, q} (π_{t} (x) g_{α} (x)) = (\int_{X} (v (x) q (f_{α} (φ (x))))^{p} d μ)^{\frac{1}{p}} ⊞ (\int_{X} (v (x) q (e^{t h (x)} g_{α} (x)))^{p} d μ)^{\frac{1}{p}} = (\int_{X} (u (x) q (f_{α} (x)))^{p} d μ)^{\frac{1}{p}} ⊞ (\int_{X} (u (x) q (g_{α} (x)))^{p} d μ)^{\frac{1}{p}} as t \to 0 = P_{u, q} (f_{α}) ⊞ P_{u, q} (g_{α}) = P_{u, q} (f_{α} ⊞ g_{α}) \to 0

(6)

This proves that (C_φ ⊞ M_{π_t}) is continuous at the origin and hence a continuous linear operator on MV₀(X, E) ⊞ MV₀(X, E).

Remark 1.2.3. Note that if φ : X → X and π_t : X → ℂ are bounded measurable complex-valued (or vector-valued) functions on X, then clearly (C_φ ⊞ M_{π_t})(f_α ⊞ g_α) is a tensor sum operator on MV₀(X) ⊞ MV₀(X) (or MV₀(X, E) ⊞ MV₀(X, E)) for any system of weights V.

If X is a system of weights generated by the characteristic functions of compact sets, then it turns out that every continuous map induces a tensor sum operator on MV₀(X) ⊞ MV₀(X) (or MV₀(X, E) ⊞ MV₀(X, E)) for any system of weights V.

Theorem 1.2.4. Let X be a completely Hausdorff space and let V = {λ_χk: λ > 0 and K ⊂ X, X is compact}.

(i) Every bounded φ : X → X and π_t : X → ℂ on MV₀(X) ⊞ MV₀(X)

(ii) Every bounded φ : X → X and π_t : X → E be an lmc with jointly continuous operator induces a tensor sum operator C_φ ⊞ M_{π_t} on MV₀(X, E) ⊞ MV₀(X, E).

Proof. Similar to the proof of Theorem 1.2.2.

Corollary 1.2.5. Let X have the discrete topology and V = {λ_χk: λ ≥ 0 and K ⊂ X, X is a finite set}. Then every function φ : X → X and π_t : X → ℂ induces a tensor sum operator (C_φ ⊞ M_{π_t})(f ⊞ g) on MV₀(X) ⊞ MV₀(X) (or MV₀(X, E) ⊞ MV₀(X, E)).

1.3. Dynamical system induced by tensor sum operators on weighted locally convex spaces of measurable functions

In this section, let us investigate the dynamical system induced by tensor sum operators on weighted locally convex spaces of measurable functions.

Theorem 1.3.1. Let U and V be arbitrary systems of weights on G and let φ : X → X and π_t : X → ℂ be continuous functions. Then C_φf ⊞ M_{π_t}g is a tensor sum operator for every t ∈ ℝ and f ⊞ g ∈ MV₀(X) ⊞ MV₀(X) iff V‖C_φ ⊞ M_{π_t}‖ ≤ U.

Proof. To show that C_φf ⊞ M_{π_t}g is a tensor sum operator, it is enough to prove that C_φf ⊞ M_{π_t}g is continuous at the origin. Let v ∈ V and B_v be a neighbourhood of the origin in MV_b(X) ⊞ MV_b(X). Then, by the given condition, there exists u ∈ U such that v‖C_φ ⊞ M_{π_t}‖ ≤ u. Now we claim that (C_φ ⊞ M_{π_t})(B_u) ⊆ B_v, where B_u is a neighbourhood of the origin in MU_b(X) ⊞ MU_b(X) (or MV_b(X, E) ⊞ MV_b(X, E)).

Let f ⊞ g ∈ B_u. Then we have

C_{φ} f ⊞ M_{π_{t}} g_{v} = ∥ C_{φ} f (x) ∥ ⊞ ∥ M_{π_{t}} g (x) ∥ = (\int_{X} (v (x) q (∥ f \circ φ (x) ∥))^{p} d μ)^{\frac{1}{p}} ⊞ (\int_{X} (v (x) q (∥ π_{t} (x) g (x) ∥))^{p} d μ)^{\frac{1}{p}} = (\int_{X} (v (x) q (∥ f (φ (x)) ∥))^{p} d μ)^{\frac{1}{p}} ⊞ (\int_{X} (v (x) q (e^{∣ t ∣∥ h ∥_{\infty}} ∥ g (x) ∥))^{p} d μ)^{\frac{1}{p}} \leq (\int_{X} (u (x) q (∥ f (x) ∥))^{p} d μ)^{\frac{1}{p}} ⊞ (\int_{X} (u (x) q (∥ g (x) ∥))^{p} d μ)^{\frac{1}{p}} \leq ∥ f (x) ⊞ g (x) ∥ \leq ∥ (f ⊞ g) (x) ∥ \leq 1.

(7)

This proves that (C_φf ⊞ M_{π_t}g) ∈ B_v and hence C_φ ⊞ M_{π_t} is a tensor sum operator.

Corollary 1.3.2. Every bounded measurable function φ : X → X and π_t : X → ℂ induces a tensor sum operator C_φ ⊞ M_{π_t} on MV_b(X) ⊞ MV_b(X) (or MV_b(X, E) ⊞ MV_b(X, E)) for a system of weights V on X.

Proof. Since C_φf ⊞ M_{π_t}g is bounded, there exists m > 0 such that |(C_φf ⊞ M_{π_t}g)(x)| ≤ m for all x ∈ X. Let v ∈ V. Then we have v(x)|C_φf ⊞ M_{π_t}g| ≤ mv(x) for all x ∈ X.

Hence, by the above theorem, C_φf ⊞ M_{π_t}g is a tensor sum operator on MV_b(X) ⊞ MV_b(X) (or MV_b(X, E) ⊞ MV_b(X, E)).

Note 1.3.3. Let h ∈ F_b(ℝ). Define π_t : ℝ → B(T) as π_t(w) = e^th(w) for all t, w ∈ ℝ.

Theorem 1.3.4. Let h ∈ F_b(ℝ). For each t ∈ ℝ, let ∇_h : ℝ × MV_b(ℝ, T) ⊞ MV_b(ℝ, T) → M(ℝ, T) ⊞ M(ℝ, T) be the function defined by ∇_h(t, f ⊞ g) = C_{φ_t} ⊞ M_{π_t}(f ⊞ g) for all t ∈ ℝ and f ⊞ g ∈ MV_b(ℝ, T) ⊞ MV_b(ℝ, T). Then ∇_h is a linear dynamical system on MV_b(ℝ, T) ⊞ MV_b(ℝ, T).

Proof. Since C_{φ_t} ⊞ M_{π_t} is a tensor sum operator on MV_b(ℝ, T) ⊞ MV_b(ℝ, T) for all t ∈ ℝ and f ⊞ g ∈ MV_b(ℝ, T) ⊞ MV_b(ℝ, T), we can conclude that ∇_h(t, f ⊞ g) ∈ MV_b(ℝ, T) ⊞ MV_b(ℝ, T) whenever t ∈ ℝ and f ⊞ g ∈ MV_b(ℝ, T) ⊞ MV_b(ℝ, T). Thus ∇_h is a function from ℝ × MV_b(ℝ, T) ⊞ MV_b(ℝ, T) into M(ℝ, T) ⊞ M(ℝ, T). It can be easily seen that ∇_h(0, f ⊞ g) = f ⊞ g and ∇_h(t + s, f ⊞ g) = ∇_h(t, ∇_h(s, f ⊞ g)).

In order to show that ∇_h is a dynamical system on MV_b(ℝ, T) ⊞ MV_b(ℝ, T), it is enough to show that ∇_h is a separately continuous map.

Let us first prove the continuity of ∇_h in the first argument. Let t_n → t. Then |t_n − t| → 0 as n → ∞. We shall show that ∇_h(t_n, f ⊞ g) → ∇_h(t, f ⊞ g) in MV_b(ℝ, T) ⊞ MV_b(ℝ, T).

Let v ∈ V. Then

P_{v} (\nabla_{h} (t_{n}, f ⊞ g) - \nabla_{h} (t, f ⊞ g))_{v} = P ((C_{φ_{t_{n}}} ⊞ M_{π_{t_{n}}}) (f ⊞ g) - (C_{φ_{t}} ⊞ M_{π_{t}}) (f ⊞ g))_{v} = ((\int_{X} (v (w) q (C_{φ_{t_{n}}} (w) f (w) ⊞ M_{π_{t_{n}}} (w) g (w)))^{p} d μ)^{\frac{1}{p}}) - ((\int_{X} (v (w) q (C_{φ_{t}} (w) f (w) ⊞ M_{π_{t}} (w) g (w)))^{p} d μ)^{\frac{1}{p}}) = ((\int_{X} (v (w) q (f (φ_{t_{n}} (w)) ⊞ π_{t_{n}} (w) g (w)))^{p} d μ)^{\frac{1}{p}}) + ((\int_{X} (v (w) q (f (φ_{t} (w)) f (w) ⊞ π_{t} (w) g (w)))^{p} d μ)^{\frac{1}{p}}) \leq ((\int_{X} (v (w) q (f (φ_{t_{n}} (w)) - e^{t_{n} h_{\infty}} g (w)))^{p} d μ)^{\frac{1}{p}}) + ((\int_{X} (v (w) q (f (φ_{t} (w)) - e^{t_{n} h_{\infty}} g (w)))^{p} d μ)^{\frac{1}{p}}) \to 0 as ∣ t_{n} - t ∣ \to 0.

(8)

Let f_α ⊞ g_α be a net in MV_b(ℝ, T) ⊞ MV_b(ℝ, T) such that f_α ⊞ g_α → f ⊞ g in MV_b(ℝ, T) ⊞ MV_b(ℝ, T). Then q(f_α ⊞ g_α − f ⊞ g)_v → 0 for all v ∈ V. We shall show that ∇_h(t, f_α ⊞ g_α) → ∇_h(t, f ⊞ g) in MV_b(ℝ, T) ⊞ MV_b(ℝ, T).

P (\nabla_{h} (t, f_{α} ⊞ g_{α}) - \nabla_{h} (t, f ⊞ g))_{v} = P (((C_{φ_{t}} ⊞ M_{π_{t}}) (f_{α} ⊞ g_{α}) (w)) - ((C_{φ_{t}} ⊞ M_{π_{t}}) (f ⊞ g) (w)))_{v} = P ((C_{φ_{t}} f_{α} ⊞ M_{π_{t}} g_{α}) - (C_{φ_{t}} f ⊞ M_{π_{t}} g))_{v} \leq ((\int_{X} (v (w) q (f_{α} (φ_{t} (w)) - f (φ_{t} (w))) e^{∣ t ∣∥ h ∥_{\infty}} q (g_{α} (w)))^{p} d μ)^{\frac{1}{p}}) + ((\int_{X} (v (w) q (f (φ_{t} (w))) [e^{∣ t ∣∥ h ∥_{\infty}} g_{α} (w) - e^{∣ t ∣∥ h ∥_{\infty}} g_{α} (w)])^{p} d μ)^{\frac{1}{p}}) \to 0 as (f_{α} ⊞ g_{α}) - (f ⊞ g) \to 0.

(9)

This proves the continuity of ∇_h, and hence ∇_h is a (linear) dynamical system on the weighted space MV_b(ℝ, T) ⊞ MV_b(ℝ, T).

1.4. Dynamical system and weighted tensor sum operator

In this section, let us investigate the dynamical system and weighted tensor sum operator.

Theorem 1.4.1. Let E be a locally convex Hausdorff space such that each convergent net in E is bounded. Let φ ∈ M(X, B(E)) and T ∈ M(X, X). Then (C_φ ⊞ M_{π_t})(f ⊞ g) is a weighted tensor sum operator on MV_b(X, E) ⊞ MV_b(X, E) iff, for every v ∈ V and p ∈ cs(E), there exist u ∈ V and q ∈ cs(E) such that v(x)P(φ(x)(w)) ≤ u(π_t(x)q(x)v for all x ∈ X and w ∈ E.

Remark 1.4.2. Let B(E) be the Banach algebra of all bounded linear operators on E. Then an operator-valued map π_t : X → B(E) is defined by π_t(x) = e^th(x) for all t ∈ ℝ and x ∈ X, where h ∈ M(X, B(E)) and ‖h‖_∞ = sup {‖h(x)‖: x ∈ X}. Also, the self-map φ_t : X → X is defined by φ_t(x) = t + x. Then we consider the weighted tensor sum operator induced by φ_t and π_t on the spaces MV₀(X, E) and MV₀(X, E).

Theorem 1.4.3. Let V be an arbitrary system of weights on X. Let ∇ : ℝ × MV_b(X, E) ⊞ MV_b(X, E) → M(X, E) ⊞ M(X, E) be the function defined by ∇(t, f ⊞ g) = (C_φ ⊞ M_{π_t})(f ⊞ g) for all t ∈ ℝ and f ⊞ g ∈ MV_b(X, E) ⊞ MV_b(X, E). Then ∇ is a linear dynamical system if, for every v ∈ V and p ∈ cs(E), there exist u ∈ V and q ∈ cs(E) such that v(x)P(φ ⊞ π_t)(x) ≤ u(π_t(x)q(x)v for all x ∈ X and w ∈ E.

Proof. For every t ∈ ℝ, C_φ ⊞ M_{π_t} is a weighted tensor sum operator on MV_b(X, E) ⊞ MV_b(X, E). Thus it follows that ∇(t, f ⊞ g) ∈ MV_b(X, E) ⊞ MV_b(X, E) for all t ∈ ℝ and f ⊞ g ∈ MV_b(X, E) ⊞ MV_b(X, E).

Clearly, ∇ is linear and

\nabla (0, f ⊞ g) (x) = (C_{φ} ⊞ M_{π_{t}}) ((f ⊞ g) (x)) for all x \in X . = (C_{φ} f ⊞ M_{π_{t}} g) (x) = f (φ (x)) ⊞ e^{t h (x)} g (x) = (f ⊞ g) (x)

(10)

Therefore, ∇(0, f ⊞ g)(x) = f ⊞ g.

Also, ∇(t + s, f ⊞ g) = ∇(t, ∇(s, f ⊞ g)).

Next, to show that ∇ is a linear dynamical system, it is sufficient to show that ∇ is a jointly continuous map [1]. Let t_n → t for all t ∈ ℝ. Then t_n − t → 0 as n → ∞. The remaining proof is similar to that of Theorem 1.3.4.

References

Chandra Kala P, Vigneswari GS. Dynamical system induced by tensor sum operator with stability theory. Int J Res Anal Rev. 2019 Mar;6(1).

Tensor Sum Operator and Dynamical System in Measurable Function Spaces

Abstract

Keywords

1. Introduction

1.1. Functions inducing tensor sum operators on weighted spaces of measurable functions

1.2. Characterisation of tensor sum operators

1.3. Dynamical system induced by tensor sum operators on weighted locally convex spaces of measurable functions

1.4. Dynamical system and weighted tensor sum operator

References

Related articles

The Hop Domination Number of Comb Product Graphs

Eccentricity Based Topological Indices of Polygonal Cylinder

Vertex Polynomial for Switching in Various Graphs

Open Access & Licensing

Related articles

The Hop Domination Number of Comb Product Graphs
D. Anusha, S. Joseph Robin
A set S ⊆ V of a graph G is a hop dominating set of G if for every v ∈ V − S, there exists u ∈ S such that d(u, v) = 2. The minimu…
03 Jun 2026

Eccentricity Based Topological Indices of Polygonal Cylinder
S. Sowmya, J. Suthiesh Goldy
A topological index plays a vital role in molecular chemistry. There are various topological descriptors in theoretical chemistry,…
19 May 2026

Vertex Polynomial for Switching in Various Graphs
P. Maya, G. R. Sanma
The vertex polynomial of a graph G is defined as V(G, x) = Σ vₖxᵏ (k = 0 to Δ(G)), where Δ(G) = max{d(v) / v ∈ V} and vₖ is the nu…
11 Feb 2026