Dim Ker Research Articles

The expression of the generalized inverse of the perturbed operator under Type I perturbation in Hilbert spaces

Let H 1, H 2 be two Hilbert spaces over the complex field C and let T: H 1 → H 2 be a bounded linear operator with the generalized inverse T +. Let T ̄ = T + δT be a bounded linear operator with ∥ T +∥ ∥ δT∥ 〈 1. Suppose that dim ker T ̄ = dim ker T 〈 ∞ or R( T ̄ ) ∩ R(T) ⊥ = 0 . Then T ̄ has the generalized inverse T +=[I−(I+T +δT) −1(I−T +T)−(I−T +)(I+T +δT) ∗−1] 1(1+δTT +) −1T +X[(I+δTT +)TT +)TT +(I+δTT +) −1+(I+δTT +) *ast;−1TT +(I+δTT +) ∗−I] −1 with ‖ T + ‖⩽ ‖T +‖ 1−‖T +‖‖δT‖ This result gives an analogue of Theorem 3.9 of M.Z. Nashed (“Generalized Inverses and Applications”, Academic Press, New York, 1976) in Hilbert spaces.

Perturbation Analysis for the Operator EquationTx = bin Banach Spaces

LetX1,X2be two Banach spaces over the complex fieldCand letT:X1→X2be a bounded linear operator with the generalized inverseT+. LetT̄=T+δTbe a bounded linear operator with ‖T+‖‖δT‖<1. Suppose thatdimkerT̄=dimkerT<∞orR(T̄)∩KerT+=0. ThenT̄has the generalized inverseT̄=(I+T+δT)−1T+with‖T̄+‖≤‖T+‖1−‖T+‖‖δT‖.This result generalizes Theorem 3.9 ofNa. Using this result, we give some results of the perturbation analysis for the operator equationTx=b.

Q-DEFORMED OSCILLATOR ALGEBRA AND AN INDEX THEOREM FOR THE PHOTON PHASE OPERATOR

The quantum deformation of the oscillator algebra and its implications on the phase operator are studied from a viewpoint of an index theorem by using an explicit matrix representation. For a positive deformation parameter q or q=exp(2πiθ) with an irrational θ, one obtains an index condition dim ker a–dim ker a†=1 which allows only a nonhermitian phase operator with dim ker eiφ–dim ker(eiφ)†=1. For q=exp(2πiθ) with a rational θ, one formally obtains the singular situation dim ker a=∞ and dim ker a†=∞, which allows a hermitian phase operator with dim ker eiΦ–dim ker(eiΦ)†=0 as well as the nonhermitian one with dim ker eiφ– dim ker(eiφ)†=1. Implications of this interpretation of the quantum deformation are discussed. We also show how to overcome the problem of negative norm for q=exp(2πiθ).

Sums and Products of Cyclic Operators

It is proved that every bounded linear operator on a complex separable Hilbert space is the sum of two cyclic operators. For the product, we show that an operator T is the product of finitely many cyclic operators if and only if the kernel of ${T^ \ast }$ is finite-dimensional. More precisely, if dim ker ${T^ \ast } \leq k(2 \leq k < \infty )$, then T is the product of at most $k + 2$ cyclic operators. We conjecture that in this case at most k cyclic operators would suffice and verify this for some special classes of operators.

A binary perfect code of length 15 and codimension 0

We construct three new binary perfect codesC 1,C 2 andC 3 of length 15. We show that dim(ker(C i))=i fori=1, 2 and 3. It follows that the codimension ofC 1 equals 0.

Perturbations of semi-Fredholm operators with complemented range

The proof of the theorem is quite involved and takes more than two pages. The purpose of this note is to give brief and transparent proofs of more general results. Our proofs rely on well known perturbation theorems which appear in [2]. It is important to note that if R(A) is complemented in Y, then R(A) is closed. This was proved in [2], IV. 1.2, without any assumptions on the kernel of A. Recall that a bounded linear operator A is called semi-Fredholm if R(A) is closed and c~(A)=dim ker A, ~(A)=codim R(A) are not both infinite. The index (A) = c~(A)-/~ (A) can be ___ ~.

Continuity of the generalized kernel and range of semi-Fredholm operators

Let X denote a Banach space over the complex field ℂ and let B(X) be the Banach algebra of all bounded linear operators on X. If T ε B(X), we write n(T) = dim ker (T) and d(T) = codim T(X). Suppose that Y is a subspace invariant under T; then TY will denote the restriction of T to Y and Y the operator on X/Y defined byY: x/Y →(Tx)/Y

Critical values of Fredholm maps

For a C r {C^r} Fredholm map A : X → Y A:X \to Y , let T k ( A ) {T_k}\left ( A \right ) be the set of x ∈ X x \in X such that codim Range D A ( x ) ≥ k DA\left ( x \right ) \geq k . The homotopy of Y − A ( T k ( A ) ) Y - A\left ( {{T_k}\left ( A \right )} \right ) is related to that of Y Y , examples are given, and a factorization result is proved.

Points of local nonconvexity, clear visibility, and starshaped sets in Rd

Let S be a compact, connected, locally starshaped set in Rd, S not convex. For every point of local nonconvexity q of S, define Aq to be the subset of S from which q is clearly visible via S. Then ker S = ∩{conv Aq: q ∈ lnc S}. Furthermore, if every d+1 points of local nonconvexity of S are clearly visible from a common d-dimensional subset of S, then dim ker S = d.

On multicyclic operators and the Vasjunin-Nikol'ski206-1206-1206-1discotheca

If A is a bounded linear multicyclic operator acting on a complex Banach spaceX, then thedisc of A is defined by: disc A = sup(R ∈ Cyc A) min{dimR′: R′ ⊂ R, R′ ∈ Cyc A}, where Cyc A denotes the family of all finite dimensional subspacesR ofX such that X = (R+AR+A 2 R+⋯)−. It is shown that if the set {λ ∈ ℂ: dim ker (λ-A)* ≥ n} has nonempty interior (in particular, if A is a Fredholm operator of index -n), then disc A ≥ n+1. This result affirmatively answers a question of V.I. Vasjunin and N.K. Nikol'skii. In the case whenX is a Hilbert space, it is shown that the set of all operators A such that A is n-multicyclic, but disc A =∞, is dense in the set of all n-multicyclic operators. If Mλ = "multiplication by λ" acting on the disk algebra (and many other spaces of continuous and/or analytic functions), then Mλ is cyclic, but disc Mλ = ∞. However, the analogous result is false if the disk algebra is replaced by the algebra of functions analytic on the disk and smooth on the boundary, or algebras of Lipschitz functions. If T is a multicyclic unicellular operator, then T is cyclic and disc T=1.

Clear Visibility and the Dimension of Kernels of Starshaped Sets

This paper will use the concept of clearly visible to obtain a Krasnosel’skii-type theorem for the dimension of the kernel of a starshaped set, and the following result will be proved: For each $k$ and $n$, $1 \leqslant k \leqslant n$, let $f(n,n) = n + 1$ and $f(n,k) = 2n$ if $1 \leqslant k \leqslant n - 1$. Let $S$ be a nonempty compact set in ${R^n}$. Then for a $k$ with $1 \leqslant k \leqslant n$, dim ker $S \geqslant k$ if and only if every $f(n,k)$ points of bdry $S$ are clearly visible from a common $k$-dimensional subset of $S$. If $k = 1$ or $k = n$, the result is best possible. Moreover, if $S$ is a compact, connected, nonconvex set in ${R^2}$, then bdry $S$ may be replaced by lnc $S$ in the theorem.

Clear visibility and the dimension of kernels of starshaped sets

This paper will use the concept of clearly visible to obtain a Krasnosel’skii-type theorem for the dimension of the kernel of a starshaped set, and the following result will be proved: For each $k$ and $n$, $1 \leqslant k \leqslant n$, let $f(n,n) = n + 1$ and $f(n,k) = 2n$ if $1 \leqslant k \leqslant n - 1$. Let $S$ be a nonempty compact set in ${R^n}$. Then for a $k$ with $1 \leqslant k \leqslant n$, dim ker $S \geqslant k$ if and only if every $f(n,k)$ points of bdry $S$ are clearly visible from a common $k$-dimensional subset of $S$. If $k = 1$ or $k = n$, the result is best possible. Moreover, if $S$ is a compact, connected, nonconvex set in ${R^2}$, then bdry $S$ may be replaced by lnc $S$ in the theorem.

K-Dimensional intersections of convex sets and convex kernels

This paper will investigate the dimension of the kernel of a compact starshaped set, and the following result will be obtained: For each k and n , 1⩽ k ⩽ n , let f(n, n) = n + 1 and f(n, k) = 2 n if 1⩽ k ⩽ n −1. Let S be a compact set in some linear topological space L . Then for a k with 1⩽ k ⩽ n , dim ker S ⩾ k if and only if for some ε > 0 and some n -dimensional flat F in L , every f(n, k) points of S see via S a common k -dimensional ge -neighborhood in F . If k = 1 or if k = n , the result is best possible. Furthermore, the proof will yield a Helly-type theorem for the dimension of intersections of compact convex sets in ℝ n .

On the spectra of finite-dimensional perturbations of matrix multiplication operators

Let Γ be a closed rectifiable curve and Ω a region in the complex plane. Suppose for each ωeΩ, R(ω) represents multiplication by an nxn-matrix of rational functions and F(ω) is a finite rank operator, both acting on the Hilbert space L 2 n (Γ). Sufficient conditions are given for the integer valued function dim ker (R(ω)+F(ω)) to be continuous at all but finitely many points in Ω. This result is applied to singular integral operators.