Displaying similar documents to “Schur Lemma and the Spectral Mapping Formula”

Subsets of nonempty joint spectrum in topological algebras

Antoni Wawrzyńczyk (2018)

Mathematica Bohemica

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We give a necessary and a sufficient condition for a subset S of a locally convex Waelbroeck algebra 𝒜 to have a non-void left joint spectrum σ l ( S ) . In particular, for a Lie subalgebra L 𝒜 we have σ l ( L ) if and only if [ L , L ] generates in 𝒜 a proper left ideal. We also obtain a version of the spectral mapping formula for a modified left joint spectrum. Analogous theorems for the right joint spectrum and the Harte spectrum are also valid.

Spectral radius of operators associated with dynamical systems in the spaces C(X)

Krzysztof Zajkowski (2005)

Banach Center Publications

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We consider operators acting in the space C(X) (X is a compact topological space) of the form A u ( x ) = ( k = 1 N e φ k T α k ) u ( x ) = k = 1 N e φ k ( x ) u ( α k ( x ) ) , u ∈ C(X), where φ k C ( X ) and α k : X X are given continuous mappings (1 ≤ k ≤ N). A new formula on the logarithm of the spectral radius r(A) is obtained. The logarithm of r(A) is defined as a nonlinear functional λ depending on the vector of functions φ = ( φ k ) k = 1 N . We prove that l n ( r ( A ) ) = λ ( φ ) = m a x ν M e s k = 1 N X φ k d ν k - λ * ( ν ) , where Mes is the set of all probability vectors of measures ν = ( ν k ) k = 1 N on X × 1,..., N and λ* is some convex lower-semicontinuous functional on...

Spectral projections for the twisted Laplacian

Herbert Koch, Fulvio Ricci (2007)

Studia Mathematica

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Let n ≥ 1, d = 2n, and let (x,y) ∈ ℝⁿ × ℝⁿ be a generic point in ℝ²ⁿ. The twisted Laplacian L = - 1 / 2 j = 1 n [ ( x j + i y j ) ² + ( y j - i x j ) ² ] has the spectrum n + 2k = λ²: k a nonnegative integer. Let P λ be the spectral projection onto the (infinite-dimensional) eigenspace. We find the optimal exponent ϱ(p) in the estimate | | P λ u | | L p ( d ) λ ϱ ( p ) | | u | | L ² ( d ) for all p ∈ [2,∞], improving previous partial results by Ratnakumar, Rawat and Thangavelu, and by Stempak and Zienkiewicz. The expression for ϱ(p) is ϱ(p) = 1/p -1/2 if 2 ≤ p ≤ 2(d+1)/(d-1), ϱ(p) = (d-2)/2 - d/p...

On the perturbation functions and similarity orbits

Haïkel Skhiri (2008)

Studia Mathematica

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We show that the essential spectral radius ϱ e ( T ) of T ∈ B(H) can be calculated by the formula ϱ e ( T ) = inf · ( X T X - 1 ) : X an invertible operator, where · ( T ) is a Φ₁-perturbation function introduced by Mbekhta [J. Operator Theory 51 (2004)]. Also, we show that if · ( T ) is a Φ₂-perturbation function [loc. cit.] and if T is a Fredholm operator, then d i s t ( 0 , σ e ( T ) ) = sup · ( X T X - 1 ) : X an invertible operator.

When spectra of lattices of z -ideals are Stone-Čech compactifications

Themba Dube (2017)

Mathematica Bohemica

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Let X be a completely regular Hausdorff space and, as usual, let C ( X ) denote the ring of real-valued continuous functions on X . The lattice of z -ideals of C ( X ) has been shown by Martínez and Zenk (2005) to be a frame. We show that the spectrum of this lattice is (homeomorphic to) β X precisely when X is a P -space. This we actually show to be true not only in spaces, but in locales as well. Recall that an ideal of a commutative ring is called a d -ideal if whenever two elements have the same annihilator...

Integral equalities for functions of unbounded spectral operators in Banach spaces

Benedetto Silvestri

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The work is dedicated to investigating a limiting procedure for extending “local” integral operator equalities to “global” ones in the sense explained below, and to applying it to obtaining generalizations of the Newton-Leibniz formula for operator-valued functions for a wide class of unbounded operators. The integral equalities considered have the form g ( R F ) f x ( R F ) d μ ( x ) = h ( R F ) . (1) They involve functions of the kind X x f x ( R F ) B ( F ) , where X is a general locally compact space, F runs over a suitable class of Banach subspaces...

Spectral synthesis and operator synthesis

K. Parthasarathy, R. Prakash (2006)

Studia Mathematica

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Relations between spectral synthesis in the Fourier algebra A(G) of a compact group G and the concept of operator synthesis due to Arveson have been studied in the literature. For an A(G)-submodule X of VN(G), X-synthesis in A(G) has been introduced by E. Kaniuth and A. Lau and studied recently by the present authors. To any such X we associate a V ( G ) -submodule X̂ of ℬ(L²(G)) (where V ( G ) is the weak-* Haagerup tensor product L ( G ) w * h L ( G ) ), define the concept of X̂-operator synthesis and prove that a...

Sufficient conditions on the existence of factors in graphs involving minimum degree

Huicai Jia, Jing Lou (2024)

Czechoslovak Mathematical Journal

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For a set { A , B , C , ... } of graphs, an { A , B , C , ... } -factor of a graph G is a spanning subgraph F of G , where each component of F is contained in { A , B , C , ... } . It is very interesting to investigate the existence of factors in a graph with given minimum degree from the prospective of eigenvalues. We first propose a tight sufficient condition in terms of the Q -spectral radius for a graph involving minimum degree to contain a star factor. Moreover, we also present tight sufficient conditions based on the Q -spectral radius...

On the spectrum of the operator which is a composition of integration and substitution

Ignat Domanov (2008)

Studia Mathematica

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Let ϕ: [0,1] → [0,1] be a nondecreasing continuous function such that ϕ(x) > x for all x ∈ (0,1). Let the operator V ϕ : f ( x ) 0 ϕ ( x ) f ( t ) d t be defined on L₂[0,1]. We prove that V ϕ has a finite number of nonzero eigenvalues if and only if ϕ(0) > 0 and ϕ(1-ε) = 1 for some 0 < ε < 1. Also, we show that the spectral trace of the operator V ϕ always equals 1.

On the convergence and character spectra of compact spaces

István Juhász, William A. R. Weiss (2010)

Fundamenta Mathematicae

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An infinite set A in a space X converges to a point p (denoted by A → p) if for every neighbourhood U of p we have |A∖U| < |A|. We call cS(p,X) = |A|: A ⊂ X and A → p the convergence spectrum of p in X and cS(X) = ⋃cS(x,X): x ∈ X the convergence spectrum of X. The character spectrum of a point p ∈ X is χS(p,X) = χ(p,Y): p is non-isolated in Y ⊂ X, and χS(X) = ⋃χS(x,X): x ∈ X is the character spectrum of X. If κ ∈ χS(p,X) for a compactum X then κ,cf(κ) ⊂ cS(p,X). A selection of our...

A note on the multiplier ideals of monomial ideals

Cheng Gong, Zhongming Tang (2015)

Czechoslovak Mathematical Journal

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Let 𝔞 [ x 1 , ... , x n ] be a monomial ideal and 𝒥 ( 𝔞 c ) the multiplier ideal of 𝔞 with coefficient c . Then 𝒥 ( 𝔞 c ) is also a monomial ideal of [ x 1 , ... , x n ] , and the equality 𝒥 ( 𝔞 c ) = 𝔞 implies that 0 < c < n + 1 . We mainly discuss the problem when 𝒥 ( 𝔞 ) = 𝔞 or 𝒥 ( 𝔞 n + 1 - ε ) = 𝔞 for all 0 < ε < 1 . It is proved that if 𝒥 ( 𝔞 ) = 𝔞 then 𝔞 is principal, and if 𝒥 ( 𝔞 n + 1 - ε ) = 𝔞 holds for all 0 < ε < 1 then 𝔞 = ( x 1 , ... , x n ) . One global result is also obtained. Let 𝔞 ˜ be the ideal sheaf on n - 1 associated with 𝔞 . Then it is proved that the equality 𝒥 ( 𝔞 ˜ ) = 𝔞 ˜ implies that 𝔞 ˜ is principal.

The strong persistence property and symbolic strong persistence property

Mehrdad Nasernejad, Kazem Khashyarmanesh, Leslie G. Roberts, Jonathan Toledo (2022)

Czechoslovak Mathematical Journal

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Let I be an ideal in a commutative Noetherian ring R . Then the ideal I has the strong persistence property if and only if ( I k + 1 : R I ) = I k for all k , and I has the symbolic strong persistence property if and only if ( I ( k + 1 ) : R I ( 1 ) ) = I ( k ) for all k , where I ( k ) denotes the k th symbolic power of I . We study the strong persistence property for some classes of monomial ideals. In particular, we present a family of primary monomial ideals failing the strong persistence property. Finally, we show that every square-free monomial...

Semi n -ideals of commutative rings

Ece Yetkin Çelikel, Hani A. Khashan (2022)

Czechoslovak Mathematical Journal

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Let R be a commutative ring with identity. A proper ideal I is said to be an n -ideal of R if for a , b R , a b I and a 0 imply b I . We give a new generalization of the concept of n -ideals by defining a proper ideal I of R to be a semi n -ideal if whenever a R is such that a 2 I , then a 0 or a I . We give some examples of semi n -ideal and investigate semi n -ideals under various contexts of constructions such as direct products, homomorphic images and localizations. We present various characterizations of this new...

Graphs with small diameter determined by their D -spectra

Ruifang Liu, Jie Xue (2018)

Czechoslovak Mathematical Journal

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Let G be a connected graph with vertex set V ( G ) = { v 1 , v 2 , ... , v n } . The distance matrix D ( G ) = ( d i j ) n × n is the matrix indexed by the vertices of G , where d i j denotes the distance between the vertices v i and v j . Suppose that λ 1 ( D ) λ 2 ( D ) λ n ( D ) are the distance spectrum of G . The graph G is said to be determined by its D -spectrum if with respect to the distance matrix D ( G ) , any graph having the same spectrum as G is isomorphic to G . We give the distance characteristic polynomial of some graphs with small diameter, and also prove that these graphs...

Simultaneous solutions of operator Sylvester equations

Sang-Gu Lee, Quoc-Phong Vu (2014)

Studia Mathematica

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We consider simultaneous solutions of operator Sylvester equations A i X - X B i = C i (1 ≤ i ≤ k), where ( A , . . . , A k ) and ( B , . . . , B k ) are commuting k-tuples of bounded linear operators on Banach spaces and ℱ, respectively, and ( C , . . . , C k ) is a (compatible) k-tuple of bounded linear operators from ℱ to , and prove that if the joint Taylor spectra of ( A , . . . , A k ) and ( B , . . . , B k ) do not intersect, then this system of Sylvester equations has a unique simultaneous solution.

( δ , 2 ) -primary ideals of a commutative ring

Gülşen Ulucak, Ece Yetkin Çelikel (2020)

Czechoslovak Mathematical Journal

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Let R be a commutative ring with nonzero identity, let ( ) be the set of all ideals of R and δ : ( ) ( ) an expansion of ideals of R defined by I δ ( I ) . We introduce the concept of ( δ , 2 ) -primary ideals in commutative rings. A proper ideal I of R is called a ( δ , 2 ) -primary ideal if whenever a , b R and a b I , then a 2 I or b 2 δ ( I ) . Our purpose is to extend the concept of 2 -ideals to ( δ , 2 ) -primary ideals of commutative rings. Then we investigate the basic properties of ( δ , 2 ) -primary ideals and also discuss the relations among ( δ , 2 ) -primary, δ -primary...

Ideals in big Lipschitz algebras of analytic functions

Thomas Vils Pedersen (2004)

Studia Mathematica

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For 0 < γ ≤ 1, let Λ γ be the big Lipschitz algebra of functions analytic on the open unit disc which satisfy a Lipschitz condition of order γ on ̅. For a closed set E on the unit circle and an inner function Q, let J γ ( E , Q ) be the closed ideal in Λ γ consisting of those functions f Λ γ for which (i) f = 0 on E, (ii) | f ( z ) - f ( w ) | = o ( | z - w | γ ) as d(z,E),d(w,E) → 0, (iii) f / Q Λ γ . Also, for a closed ideal I in Λ γ , let E I = z ∈ : f(z) = 0 for every f ∈ I and let Q I be the greatest common divisor of the inner parts of non-zero functions...

Truncated spectral regularization for an ill-posed non-linear parabolic problem

Ajoy Jana, M. Thamban Nair (2019)

Czechoslovak Mathematical Journal

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It is known that the nonlinear nonhomogeneous backward Cauchy problem u t ( t ) + A u ( t ) = f ( t , u ( t ) ) , 0 t < τ with u ( τ ) = φ , where A is a densely defined positive self-adjoint unbounded operator on a Hilbert space, is ill-posed in the sense that small perturbations in the final value can lead to large deviations in the solution. We show, under suitable conditions on φ and f , that a solution of the above problem satisfies an integral equation involving the spectral representation of A , which is also ill-posed. Spectral truncation...

On the regularity and defect sequence of monomial and binomial ideals

Keivan Borna, Abolfazl Mohajer (2019)

Czechoslovak Mathematical Journal

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When S is a polynomial ring or more generally a standard graded algebra over a field K , with homogeneous maximal ideal 𝔪 , it is known that for an ideal I of S , the regularity of powers of I becomes eventually a linear function, i.e., reg ( I m ) = d m + e for m 0 and some integers d , e . This motivates writing reg ( I m ) = d m + e m for every m 0 . The sequence e m , called the of the ideal I , is the subject of much research and its nature is still widely unexplored. We know that e m is eventually constant. In this article, after proving...

On the mappings 𝒵 A and A in intermediate rings of C ( X )

Mehdi Parsinia (2018)

Commentationes Mathematicae Universitatis Carolinae

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In this article, we investigate new topological descriptions for two well-known mappings 𝒵 A and A defined on intermediate rings A ( X ) of C ( X ) . Using this, coincidence of each two classes of z -ideals, 𝒵 A -ideals and A -ideals of A ( X ) is studied. Moreover, we answer five questions concerning the mapping A raised in [J. Sack, S. Watson, C and C * among intermediate rings, Topology Proc. 43 (2014), 69–82].