In the present paper, we introduce two mean transforms and describe them in the context of the weighted composition operators. We also discuss various structural properties of these mean transforms and compare them with some known transforms, namely polar decomposition, Aluthge transform, Duggal transform, and mean transform.

Peter Lappan in [9] proved that for each n∈ℕ={1,2,3,…}, let f_1,n, f_2,n and f_3,n be three continuous functions on 𝔻:={z∈ℂ : |z| < 1} such that for each j=1,2,3, the sequence (f_j,n) converges locally uniformly to a function f_j on 𝔻. Suppose that the three functions f_1, f_2, and f_3 avoid each other on 𝔻. Let ℱ =(g_n) be a sequence of meromorphic functions in 𝔻 with the property that for each n, the four functions g_n, f_1,n, f_2,n, and f_3,n avoid each other, then ℱ is normal. We present here an analogue of this result in the setting of quasiregular mappings. We also obtain analogues of a few other results by Peter Lappan in [9] to quasiregular setting in the Euclidean space ℝ^n for normal families and normal quasiregular mappings.

In this paper, the notion of compression of kth order slant Hankel operator to model space is introduced. Also, we show a connection between compression of kth order slant Toeplitz operators and that of Hankel operators. Using these connections, we provide various characterizations for an operator to be of this type. In addition, we characterize these operators in terms of operators of rank at most k+1 .

For an n-tuple, λ:= (λ _1, λ _2, … , λ _n) in ℂ^n, we introduce the k th-order λ -slant Hankel operators on the Lebesgue space L^2(𝕋^n) , where 𝕋 is the unit circle and k:= (k_1,k_2, … ,k_n) is an n-tuple with each k_i ≥ 1, an integer ( 1 ≤ i ≤ n ). These operators are described in terms of solutions of a system of operator equations. Further we study these operators with reference to the Calkin algebra.

The aim of this article is to detect the ascent and descent of weighted composition operators on Lorentz spaces. We investigate the conditions on a measurable transformation T and a complex-valued measurable function u defined on a measure space (X,𝒜,μ ) that cause the weighted composition operators on Lorentz space L(p, q), 1

In this paper, we discuss Asymmetric Truncated Hankel Operators (ATHO) involving the reflecting operator used in the classical definition of Hankel operators, on a model space. A characterization is obtained for the symbol to induce the zero ATHO. Further, some connections are established between an asymmetric truncated Toeplitz operator and an asymmetric truncated Hankel operator using conjugations.

An extension of slant Hankel operator,namely,the kth-order λ-slant Hankel operator on the Lebesgue space L2(Tn),where T is the unit circle and n ≥ 1,a natural number,is described in terms of the solution of a system of operator equations,which is subsequently expressed in terms of the product of a slant Hankel operator and a unitary operator.The study is further lifted in Calkin algebra in terms of essentially kth-order λ-slant Hankel operators on L2(Tn).

The paper focuses on determining the ascent and descent of multiplication, composition, and weighted composition operators on variable exponent Lebesgue spaces. We explore the conditions on the measurable functions u and measurable transformations T defined on σ - finite complete measure space (X,𝒜,μ ) that cause these operators on variable exponent Lebesgue spaces to have finite or infinite ascent (descent).

In the paper, we introduce the notion of compression of generalized slant Toeplitz operators to the Hardy space of n-dimensional torus T-n. It deals with characterizations of introduced operator with specific as well as general symbols. Certain algebraic and structural properties of considered operators are also investigated. Finally, we discuss few results related to essentially kth-order lambda-slant Toeplitz operator.

We study composition operators between variable exponent Lebesgue spaces and characterize boundedness and compactness of the composition operators on a variable exponent Lebesgue space. We also derive a sufficient condition for composition operator to have a closed range and explain some properties which these operators share with the case of Lebesgue spaces.

In this paper, we characterize the boundedness of weighted composition operators, induced by measurable transformations and complex-valued measurable functions, on variable exponent Lebesgue spaces. We also derive conditions for these operators to be compact or injective or have closed range. In addition, we investigate some relations between these operators and multiplication operators.

We consider Hankel and Toeplitz operators on H 2 ( T n ), the Hardy space of the n- torus T n . Given symbols phi and psi in L infinity ( T n ) with suitable properties, we obtain necessary and sufficient conditions for the Hankel operator H psi,n and the Toeplitz operator T phi,n to commute. We then extend the study to the more general situation where no assumptions are imposed on phi, and provide new, non-trivial necessary conditions for the commutativity of H psi,n and T phi,n . We also show that certain well known commutativity results between Hankel and Toeplitz operators in the one-variable case do not extend to the multivariable setting. (c) 2024 Elsevier Masson SAS. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Let 𝕌m ⊂ ℂm be a unit ball centered at the origin and let ℙn be an n-dimensional complex projective space with the metric Eℙn. Moreover, let φ: [0, 1) → (0,∞) be a smoothly increasing function. A holomorphic mapping f : 𝕌m → ℙn is called φ-normal if (φ||z||))−1(Eℙn(f(z), df (z))(ξ)) is bounded above for z ∈ 𝕌m and ξ ∈ ℂm such that ||ξ|| = 1, where df (z) is a map from Tz(𝕌m) into Tf(z) (ℙn) induced by f. For n = 1, f is called a φ-normal function. We present an extension of Lappan’s five-valued theorem to the class of φ-normal functions.

In this paper the notion of slantification of a Hankel operator on the space H^2(𝕋^n) , the Hardy space of n -torus, is introduced. Various properties including hyponormality, isometric behaviour, co-isometric behaviour and compactness of these operators are also studied.

We introduce a k th-order slant Hankel operator on the Lebesgue space of n -torus, for k = (k_1,… ,k_n) , where each k_t ≥ 1, (1 ≤ t ≤ n) is an integer. Our main result is to obtain equivalent characterizations for a bounded operator on L^2(𝕋^n) to be a k th-order slant Hankel operator. We also discuss various commutative, spectral and other properties of these operators.

The goal of this research is to discuss the ascent (descent) and essential ascent (essential descent) for weighted composition operators on Lorentz sequence spaces. We look into the conditions on self-map T and the complex valued function u defined on the set of natural numbers so that the induced weighted composition operators on Lorentz sequence space l(p, q), 1 < p <= infinity, 1 <= q <= infinity to have finite or infinite ascent (descent, essential ascent or essential descent). Our findings are well supported by a number of examples. (C) 2022 Elsevier Inc. All rights reserved.

In this paper, the commutative and spectral properties of a kth-order slant Hankel operator (k ≥ 2, a fixed integer) on the Lebesgue space of n-dimensional torus, Tn, where T is the unit circle, are studied. Characterizations for the commutativity and essential commutativity between higher order slant Hankel operators and slant Toeplitz operators have been obtained. The presence of an open disk in the point spectrum of a kth-order slant Hankel operator with a unimodular inducing function has also been ensured.

In this paper, we provide various characterizations for the composition operator on Lorentz spaces L(p, q), 1 < p <= infinity, 1 <= q <= infinity to have finite ascent (descent) in terms of its inducing measurable transformation. At the end, in order to demonstrate our outcomes, some examples are given.

The paper introduces the [Formula: see text]th-order slant Toeplitz operator on the Lebesgue space of [Formula: see text]-torus, where [Formula: see text] such that [Formula: see text] for all [Formula: see text]. It investigates certain properties of [Formula: see text]th-order slant Toeplitz operators on the Lebesgue space [Formula: see text]. The paper deals with a system of operator equations, characterizing the [Formula: see text]th-order slant Toeplitz operators. At the end, we discuss certain spectral properties of the considered operator.

In the present paper, properties of the self map on $$\mathbb {D}$$ , the open unit disc, and complex valued mapping on $$\mathbb {D}$$ are obtained when the induced k th-order weighted generalized composition operator on weighted Hardy space is Fredholm. We also describe the action of the adjoint of generalized multiplication, generalized composition and $$k{th}$$ -order weighted generalized composition operators on the derivative evaluation kernel functions, which in turn provide some invariant subspaces for these operators.