Introduction to Functional Analysis.

A hilbert space problem book

This is lecture notes for several courses on Functional Analysis at School of Mathematics of University of Leeds. They are based on the notes of Dr. Matt Daws, Prof. Jonathan R. Partington and Dr. David Salinger used in the previous years. Some sections are borrowed from the textbooks, which I used since being a student myself. However all misprints, omissions, and errors are only my responsibility. I am very grateful to Filipa Soares de Almeida, Eric Borgnet, Pasc Gavruta for pointing out some of them. Please let me know if you find more. The notes are available also for download in PDF. The suggested textbooks are [1,6,8,9]. The other nice books with many interesting problems are [3, 7]. Exercises with stars are not a part of mandatory material but are nevertheless worth to hear about. And they are not necessarily difficult, try to solve them! CONTENTS List of Figures 3 Notations and Assumptions 4 Integrability conditions 4 1. Motivating Example: Fourier Series 4 1.1. Fourier series: basic notions 4 1.2. The vibrating string 8 1.3. Historic: Joseph Fourier 10 2. Basics of Linear Spaces 11 2.1. Banach spaces (basic definitions only) 12 2.2. Hilbert spaces 14 2.3. Subspaces 16 2.4. Linear spans 19 3. Orthogonality 20 3.1. Orthogonal System in Hilbert Space 21 3.2. Bessel’s inequality 23 3.3. The Riesz–Fischer theorem 25 3.4. Construction of Orthonormal Sequences 26 3.5. Orthogonal complements 28 4. Fourier Analysis 29 Date: 16th October 2017. 1

Introduction to Functional Analysis. By Angus E. Taylor. Pp. xvi. 423. $12.50. 1958. (John Wiley and Sons)

Let T be a bounded linear operator on a complex Hilbert space H. In this paper we introduce a new class, denoted KQA∗, of operators satisfying T ∗k(|T | − |T ∗|2)T k ≥ 0 where k is a natural number, and we prove basic structural properties of these operators. Using these results, we also show that if E is the Riesz idempotent for a non-zero isolated point μ of the spectrum of T ∈ KQA∗, then E is self-adjoint and EH = ker(T − μ) = ker (T − μ)∗. Some spectral properties are also presented.

Isolated points of spectrum of $k$-quasi-$*$-class $A$ operators

In the present article, we introduce a new class of operators which will be called the class of k-quasi ∗-paranormal operators that includes ∗-paranormal operators. A part from other results, we show that following results hold for a k-quasi ∗-paranormal operator T: (i) T has the SVEP. (ii) Every non-zero isolated point in the spectrum of T is a simple pole of the resolvent of T. (iii) All Weyl type theorems hold for T. (iv) Comments and some open problems are also presented.

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On a new class of operators and Weyl type theorems

On ∗-paranormal contractions and properties for ∗-class A operators

Given Hilbert space operators $T, S\in\B$, let $\triangle$ and $\delta\in B(\B)$ denote the elementary operators $\triangle_{T,S}(X)=(L_TR_S-I)(X)=TXS-X$ and $\delta_{T,S}(X)=(L_T-R_S)(X)=TX-XS$. Let $d=\triangle$ or $\delta$. Assuming $T$ commutes with $S^*$, and choosing $X$ to be the positive operator $S^{*n}S^n$ for some positive integer $n$, this paper exploits properties of elementary operators to study the structure of $n$-quasi $[m,d]$-operators $d^m_{T,S}(X)=0$ to bring together, and improve upon, extant results for a number of classes of operators, amongst them $n$-quasi left $m$-invertible operators, $n$-quasi $m$-isometric operators, $n$-quasi $m$-selfadjoint operators and $n$-quasi $(m,C)$ symmetric operators (for some conjugation $C$ of $\H$). It is proved that $S^n$ is the perturbation by a nilpotent of the direct sum of an operator $S_1^n=(S|_{\overline{S^n(\H)}})^n$ satisfying $d^m_{T_1,S_1}(I_1)=0$, $T_1=T|_{\overline{S^n(\H)}}$, with the $0$ operator; if also $S$ is left invertible, then $S^n$ is similar to an operator $B$ such that $d^m_{B^*,B}(I)=0$. For power bounded $S$ and $T$ such that $ST^*-T^*S=0$ and $\triangle_{T,S}(S^{*n}S^n)=0$, $S$ is polaroid (i.e., isolated points of the spectrum are poles). The product property, and the perturbation by a commuting nilpotent property, of operators $T, S$ satisfying $d^m_{T,S}(I)=0$, given certain commutativity properties, transfers to operators satisfying $S^{*n}d^m_{T,S}(I)S^n=0$.

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Structure of n-quasi left m-invertible and related classes of operators

Generalized browder, weyl spectra and the polaroid property under compact perturbations

Given a Banach space X ,l etMC ∈ B(X ⊕ X ) denote the upper triangular operator matrix MC = ( AC B ) ,a nd letδAB ∈ B(B(X )) denote the generalized derivation δAB(X )= AX - XB .I f lim n→∞ � δ n (C)� 1 n =0 , thenσx(MC )= σx(M0), where σx stands for the spectrum or a distinguished part thereof (but not the point spectrum); furthermore, if R = R1 ⊕ R2 ∈ B(X ⊕ X ) is a Riesz operator which commutes with MC, then σx(MC + R )= σx(MC), where σx stands for the Fredholm essential spectrum or a distinguished part thereof. These results are applied to prove the equivalence of Browder's (a-Browder's) theorem for M0, MC, M0 + R and MC + R. Sufficient conditions for the equivalence of Weyl's (a-Weyl's) theorem are also considered. MSC: Primary 47B40; 47A10; secondary 47B47; 47A11

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In Hyoun Kim

Papers

On ∗-paranormal contractions and properties for ∗-class A operators

Structure of n-quasi left m-invertible and related classes of operators

Generalized browder, weyl spectra and the polaroid property under compact perturbations

Upper triangular operator matrices, asymptotic intertwining and Browder, Weyl theorems

Structure of $n$-quasi left $m$-invertible and related classes of operators