Mehdi Radjabalipour

The Academy of Sciences of Islamic Republic of Iran | IAS · Department of Mathematics

Let $N\in B(H)$ be a normal operator acting on a real or complex Hilbert space $H$‎. ‎Define $N^\dagger:=N_1^{-1}\oplus 0:\mathcal{R}(N)\oplus \mathcal{K}(N)\rightarrow H$‎, ‎where $N_1=N|_{\mathcal{R}(N)}$‎. ‎Let the {\it fractional semigroup} $\mathfrak{F}r(W)$ denote the collection of all words of the form $f_1^\diamond f_2^\diamond \cdots f_k^\...

In spite of the important applications of real selfadjoint operators and monotone operators, very few papers have dealt in depth with the properties of such operators. In the present paper, we follow A. Rhodius to define the spectrum \(\sigma _{\mathbb {F}}(T)\) and the numerical range \(W_{\mathbb {F}}(T)\) of a selfadjoint operator T acting on a...

Given a densely defined closed operator T:D(T)⊂H→K, von Neumann defined W:=(I+T∗T)-1 and showed that 0≤W≤I, K(W)={0} and T∗T=(I-W)W-1=W-1(I-W) with D(T∗T)=R(W). (Here, D(·), R(·), K(·) and, later, G(T) stand for the domain, the range, the kernel, and the graph of a linear transformation, respectively.) The functional calculus is not applicable, in...

Solving linear system of differential equations by Jordan canonical form needs the change in the real field to complex and then retrieve the complex solutions to the real ones. B. Malesevic, D. Todoric, I. Jovovic, and S. Telebakovic suggest that it is more convenient to apply the rational canonical form than the Jordan canonical form. They reduce...

Let Bs(H) denote the set of all bounded selfadjoint operators acting on a separable complex Hilbert space H of dimension ≥2. Also, let SAs(H) (esp. IAs(H)) denote the class of all singular (resp. invertible) algebraic operators in Bs(H). Assume Φ:Bs(H)→Bs(H) is a unital additive surjective map such that Φ(SAs(H))=SAs(H) (resp. Φ(IAs(H))=IAs(H)). Th...

Necessary and sufficient conditions are obtained for a sequence {xj: j ϵ J} in a Hilbert space to be, up to the elimination of a finite subset of J, the linear homeomorphic image of an orthogonal basis of some Hilbert space K. This extends a similar result for orthonormal bases due to Holub [J.R. Holub. Pre-frame operators, Besselian frames, and ne...

For a (finite or infinite dimensional) vector space V, the notion of a symmetric Jordan canonical form of an operator having a minimal polynomial is defined and used to verify the relation between the notions of “Jordan canonical form” and “rational canonical form.” The paper extends and repairs Theorem 2.2 of [M. Radjabalipour, The rational canoni...

The famous primary and cyclic decomposition theorems along with the tightly related rational and Jordan canonical forms are extended to linear spaces of infinite dimensions with counterexamples showing the scope of extensions.

It is clear that a given rational canonical form can be further resolved to a Jordan canonical form with entries from the splitting field of its minimal polynomial. Conversely, with an a priori knowledge of the existence and uniqueness of the rational canonical form of a matrix with entries from a general field, one can modify its Jordan canonical...

It is easy to see that if $\cG$ is a non-abelian group of unitary matrices, then for no members $A$ and $B$ of $\cG$ can the rank of $AB-BA$ be one. We examine the consequences of the assumption that this rank is at most two for a general semigroup $\cS$ of linear operators. Our conclusion is that under obviously necessary, but trivial, size condit...

Algebraic frames are generalizations of Fourier transforms on locally compact abelian groups in the sense that the family of vectors forming the frame are replaced by a family of unbounded linear functionals. The paper studies the indexing measure space of the algebraic frames; as the investigation narrows down to the class of lower semi-frames, mo...

Bauschke, Borwein and Wang have shown in [H.H. Bauschke, J.M. Borwein, X. Wang, Fitzpatrick functions and continuous linear monotone operators, Siam J. Optimization, 18 (3) (2007), 789-809] that if FT({dotoperator},{dotoperator}) denotes the Fitzpatrick function of a continuous linear monotone operator T on a separable real Hilbert space H, then .F...

The paper studies bounded or unbounded operators which can act as analysis operators or synthesis operators of various signal processing including generalized frames, semi-frames, discrete frames, Fourier transforms, etc. The paper is concluded by a short discussion of the controllability of the behavior of the processed signals.

The main purpose of the Note is to show that if the second Aluthge transform of an invertible operator is normal, so it is its first Aluthge transform. This extends results due to Moslehian and Nabavi Sales [Some conditions implying normality of operators, C. R. Math. Acad. Sci. Paris, Ser. I 349 (2011) 251–254] and Rose and Spitkovsky [On the stab...

Let { m : m ∈ M} be a generalized frame in Hilbert space H with frame bounds 0 < A ≤ B B < ∞ and the analysis operator T : H → L 2(μ). The paper studies the relation between (space) redundancy (TH) ⊥ and (norm) redundancy A. Also, in case dimH < ∞, the effect of the redundancies on the reduction of the total energy of noise is studied.

Let X be a real or complex locally convex vector space and Lc(X) denote the ring (in fact the algebra) of continuous linear operators on X. In this note, we characterize certain one-sided ideals of the ring Lc(X) in terms of their rank-one idempotents. We use our main result to show that a one-sided ideal of the ring of continuous linear operators...

For X,Y ∈ Mnm(R )( =Mnm), we say that Y is left (resp. right) matrix majorized by X and write Y ≺� X (resp. Y ≺r X )i fY = RX (resp. Y = XR) for some row stochastic matrix R. A linear operator T:Mnm → Mnm is said to be a linear preserver of a given relation ≺ on Mnm if Y ≺ X implies that TY ≺ TX. The linear preservers of ≺� or ≺r are fully characte...

Let D be a division ring, V a right or left vector space over D, and L(V) the ring of all right (resp. left) linear transformations on V. We characterize certain one-sided ideals of the ring L( V) in terms of their rank-one idempotents. We use our result to characterize a division ring D in terms of the one-sided ideals of M-n(D). Some other conseq...

An n × m matrix A is said to be matrix majorized (or more precisely matrix majorized from the right) by an n × m matrix B, and write A ≺ B, if there exists a row stochastic matrix R such that A = BR. We characterize the linear operators that preserve the matrix majorization ≺.

For X,Y 2 Mnm(:= Mnm(R)), we say X is weakly matrix majorized or matrix majorized from the left by Y and write X ‘ Y , if X = RY for some row stochastic matrix R. Also we write X,‘ Y if X,‘ Y,‘ X. A mapping,T : Mnm,! Mnm,is said to be a strong preserver of ‘, if {X 2 Mnm : X ‘ A} = {X 2 Mnm,: TX,‘ TA} for all A 2 Mnm. Two such strong preservers T a...

A matrix majorization relation A ≺r B (resp., A ≺� B) on the collection Mn of all n × n real matrices is a relation A = BR (resp., A = RB )f or somen × n row stochastic ma- trix R (depending on A and B). These right and left matrix majorizations have been considered by some authors under the names "matrix majorization" and "weak matrix majorization...

The main purpose of this paper is to characterize triangularizable matrices A∈Mn(F) whose commutants are triangularizable, where F is an arbitrary field. More precisely, we show that the commutant of a triangularizable matrix A∈Mn(F) is triangularizable if and only if for any eigenvalue λ of A, the corresponding Jordan blocks in the Jordan canonica...

Let be a subring of Mn(D) for some division ring D satisfying the following three conditions: (i) there exists a division subring K of D such that for all a ∈ K and all ; (ii) for every , there exist ; and (iii) A = 0 if . It is shown that contains a maximal central idempotent C such that , and if E in in the center of are minimal and is a division...

Let x=g(t,x(t),u(t)) be the governing equation of an optimal control problem with two-point boundary conditions h 0(x(a))+h 1(x(b)) = 0, where x: [a,b] → ℝn is continuous, u: [a,b] → ℝk-n is piecewise continuous and left continuous, h0,h1: ℝn → ℝq are continuously differentiable, and g:[a,b]× ℝk → ℝn is continuous. The paper finds functions ξ i ∈ C...

A version of Burnside's theorem states that if F is an arbitrary field and A⊂Mn(F) is an irreducible (or, equivalently, transitive) subalgebra containing a rank-one matrix, then A=Mn(F). The present paper shows that if F is replaced by a division ring D, then every transitive left subalgebra of Mn(D) containing a rank-one matrix is equal to Mn(D)....

A version of Burnside's theorem states that if F is an arbitrary field and is an irreducible (or, equivalently, transitive) subalgebra containing a rank-one matrix, then . The present paper shows that if F is replaced by a division ring D, then every transitive left subalgebra of Mn(D) containing a rank-one matrix is equal to Mn(D). (Here, by a lef...

Given Hilbert spaces H and K and a von Neumann algebra A⊂B(H), let Φ denote the class of all additive mappings ϕ:A→B(K) satisfying |ϕ(A)|=ϕ(|A|) (A∈A). The paper shows that if A contains no nonzero abelian central projection then every ϕ∈Φ preserves the *-operation, the R-linear combination, and, up to a commuting operator multiple ϕ(I)⩾0, the (rin...

In this paper a simplified quasi-two-dimensional model for small signal gain optimization in gasdynamic laser is introduced. In order to obtain a homogeneous medium with maximum optical gain in the active medium, by nozzle shape formation, the shock occurrence position is controlled and is postponed to some point behind the laser active medium. The...

We construct discrete frames (wavelets) associated to generalized frames (wavelets) with an emphasis on those associated with windowed Fourier transforms.

It is shown that an additive map ϕ:B(H)→B(K) is the sum of two *-homomorphisms, one of which is C-linear and the other is C-antilinear provided that 1.[(a)] |ϕ(A)|=ϕ(|A|) for all A∈B(H),2.[(b)] ϕ(I) is an orthogonal projection, and3.[(c)] ϕ(iI)K⊂ϕ(I)K. The structure of ϕ is more refined when it is injective. The paper also studies the properties of...

Let h be a generalized frame in a separable Hilbert space H in- dexed by a measure space (M;S;), and assume its analysing operator is surjective. It is shown that h is essentially discrete; that is, the correspond- ing index measure space (M;S;) can be decomposed into atoms E1;E2; such that L2() is isometrically isomorphic to the weighted space '2w...

Let h h be a generalized frame in a separable Hilbert space H H indexed by a measure space ( M , S , μ ) (M,\mathcal { S},\mu ) , and assume its analysing operator is surjective. It is shown that h h is essentially discrete; that is, the corresponding index measure space ( M , S , μ ) (M,\mathcal { S},\mu ) can be decomposed into atoms E 1 , E 2 ,...

In this paper, the method of calculus of variation is used for finding the supersonic part of the nozzle of a gasdynamic laser with optimal gain in which Mach lines intersect at some points outside the active medium. This will control the position of shock occurrence and also will ensure the optical uniformity of the active medium. The interesting...

A technical lemma is proved for certain semigroups of matrices. It has several applications to problems concerning irreducible semigroups satisfying spectral conditions, e.g., submultiplicativity of spectrum. It is also used to give extensions of the following theorem of Brauer's. If U is a finite group of complex matrices, so that for some integer...

Following the terminology used by Gohberg, Lancaster, and Rodman, the main results of the paper are as follows. (i) Studying the values of the partial multiplicities of a matrix polynomial A(λ) = λ2 1 + λC + K with hermitian coefficients at real eigenvalues λ0 and determining sharp bounds for the highest degree d of the factor (λ - λo)d in the biva...

Families A and B of n X n complex matrices with bounded products are studied. In particular, it is shown that if A = B-k := {B1B2 ... B-k : B-i epsilon B (i = 1, 2,..., k)} is bounded for some k, then B-m is bounded for all m greater than or equal to n. The latter result is used to extend the relation lim sup ($) over cap p(k)(B)(1/k)less than or e...

Among other results, it is shown that ifC andK are arbitrary complexn×n matrices and if det(λ 02 Iλ0C+K)=0 for some λ0≠0 (resp. λ0=0), then the Newton diagram of the polynomialt(λ, ε) = det(λ2I+λ(1+ε)C+K expanded in (λ−λ0) and ε, has at least a point on or below the linex+y=b (resp. has no expanded in (λ−λ0) and ε, has at least a point on or below...

Without Abstract

It is shown that an irreducible semigroup of n × n complex matrices with real spectra is simultaneously similar to a semigroup of real matrices. Weaker results are obtained for semigroups of matrices over a general field with traces in a subfield.

Using Scott Brown's techniques, J. Eschmeier and B. Prunaru showed that if T is the restriction of a decomposable (or S-decomposable) operator B to an invariant subspace such that (T) is dominating in C/S for some closed set S, then T has an invariant subspace. In the present paper we prove various invariant subspace theorems by weakening the decom...

A characterization of spectral operators due to N. Dunford is simplified. Especially, his complicated Condition (D) is replaced by a very simple one.

If A is a norm closed algebra of compact operators on a Hilbert space and if its Jacobson radical J(A) consists of all quasinilpotent operators in A then A/ J(A) is commutative. The result is not valid for a general algebra of polynomially compact operators.

The minimal projections of a transitive algebra of n × n matrices are studied. The result is then applied to find new criteria for such an algebra to be equal to the whole algebra of n × n matrices. Also, triangularizability of an algebra satisfying tr(ABC) = tr(BAC) for all A,B,C in the algebra is examined in terms of its minimal projections.

A sufficient condition is obtained for two isometries to be unitarily equivalent. Also, a new class of M-hyponormal operator is constructed.

A simple proof of the previously known fact that an operator is decomposable if its adjoint is decomposable is given.

Soit N un operateur normal qui est une transformee quasi affine d'un operateur normal M. On montre que si M (resp. N) est la somme directe de k (resp. m) copies d'un operateur A ayant un commuttant commutatif, ou m et k sont des cardinalites denombrables, et si N est un transforme quasi affine de M, alors k=m

An algebra of operators having the property of the title is constructed and it is used to give examples related to some recent invariant subspace results.

For each natural number n we define to be the class of all weakly closed algebras of (bounded linear) operators on a separable Hilbert space H such that the lattice of invariant subspaces of and (alg lat ) ( n ) are the same. (If A is an operator, A ( n ) denotes the direct sum of n copies of A; if is a collection of operators, . Also, alg lat deno...

Necessary and sufficient conditions are obtained for an operator to commute with a positive operator.

Necessary and sufficient conditions are obtained for an operator to commute with a positive operator.

A matricial representation is given for the algebra of operators leaving a given dense operator range invariant. It is shown that every operator on an infinite-dimensional Hilbert space has an uncountable family of invariant operator ranges, any two of which intersect only in (0).

It is shown that every 2-decomposable operator (in the sense of S. Plafker) is decomposable (in the sense of C. Foias); this answers a question raised by Plafker. © 1978, University of California, Berkeley. All Rights Reserved.

In this paper we first study the equality of two operators whose values at each point satisfy certain inequalities, and then, somehow related, we examine the possibility of writing certain operators as products of two self-ad joint operators.

Recent results of C. R. Putnam are used to find some conditions for normality of operators. The emphasis is on the classes of spectral operators (defined by N. Dunford) and $M$-hyponormal operators (defined by J. G. Stampfli).

Recent results of C. R. Putnam are used to find some conditions for normality of operators. The emphasis is on the classes of spectral operators (defined by N. Dunford) and W-hyponormal operators (defined by J. G. Stampfli).

A bounded convex set G in the plane is the numerical range of an operator on a separable Hilbert space if G\G0 is a countable union of arcs of conic sections and singletons. This result answers, in particular, a question raised by Joel Anderson.

Let A A be a Hilbert-space operator satisfying the growth condition | | ( z − A ) − 1 | | ⩽ exp ⁡ { K [ dist ⁡ ( z , J ) ] − S } , z ∉ J ||{(z - A)^{ - 1}}|| \leqslant \exp \{ K{[\operatorname {dist} (z,\;J)]^{ - S}}\} ,\;z \notin J , where J J is a C 2 {C^2} Jordan curve, and K > 0 , s ϵ ( 0 , 1 ) K > 0,\;s\epsilon (0,\;1) are two constants. Let T...

Let A be a Hilbert-space operator satisfying the growth condition $\|(z - A)^{-1}\| \leq\le \exp\{K \lbrack\operatorname{dist} (z, J) \rbrack^{-s} \}, z \not\in J$, where J is a C2 Jordan curve, and $K > 0, s \in (0, 1)$ are two constants. Let T = A + B for some $B \epsilon C_p, 1 \leq\le p

Let T be the adjoint of a subnormal operator defined on a Hilbert space H . For any closed set δ \delta , let X T ( δ ) = { x ∈ H {X_T}(\delta ) = \{ x \in H : there exists an analytic function f x : C ∖ δ → H {f_x}:{\text {C}}\backslash \delta \to H such that ( z − T ) f x ( z ) ≡ x } (z - T){f_x}(z) \equiv x\} . It is shown that T is decomposable...

The main purpose of this paper is to show that a bounded Hilbert-space operator whose imaginary part is in the Schatten class C p (1 ≦ p < ∞ ) is strongly decomposable. This answers affirmatively a question raised by Colojoara and Foias [ 6 , Section 5(e), p. 218]. In case 0 ≦ T* — T ∈ C 1 , it was shown by B. Sz.-Nagy and C. Foias [ 2 , p. 442; 25...

Throughout this paper T will denote a bounded linear operator which is defined on a Banach space and whose spectrum lies on a rectifiable Jordan curve J . The operators having some growth conditions on their resolvents have been the subject of discussion for a long time. Many sufficient conditions have been found to ensure that such operators have...

In an axisymmetric CO 2 -N 2 -H 2 O gas dynamic laser, let Γ denote the intersection of the vertical plane of symmetry with the upper part of the (supersonic) nozzle. To obtain a maximal small signal gain, some authors have tested several families of curves for Γ. To find the most general solution for Γ, an application of Pontryagin's principle led...

The paper investigates the problems that students usually have when beginning the study of the notion of limit in its understanding and usage.

Let B(H) and B(K) denote the algebras of all (bounded linear) operators on Hilbert spaces H and K, respectively. It is shown that an additive map φ:B(H)→B(K) is the sum of two *-homomorphisms one of which C-linear and the other C-antilinear provided that (a) φ(I) is an orthogonal projection, (b) φ(iI)K⊂φ(I)K, and (c) |φ(A)|=φ(|A|) for all A∈B(H). T...

Vita. Thesis (Ph. D.)--University of Toronto, 1973. Includes bibliographical references (leaves [80-84]). Microfiche of typescript.