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Approximation Properties and Eigenvalue Problems for Banach Spaces View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Pakistan Research Repository Approximation properties and eigenvalue problems for Banach spaces Qaisar Latif February 12, 2015 Approximation properties and eigenvalue problems for Banach spaces Name: Qaisar Latif Year of Admission: 2009 Registration No.: 105{GCU{PHD{SMS{09 Abdus Salam School of Mathematical Sciences GC University, Lahore, Pakistan. Approximation properties and eigenvalue problems for Banach spaces Submitted to Abdus Salam School of Mathematical Sciences GC University Lahore, Pakistan in the partial fulfillment of the requirements for the award of degree of Doctor of Philosophy in Mathematics By Name: Qaisar Latif Year of Admission: 2009 Registration No.: 105{GCU{PHD{SMS{09 Abdus Salam School of Mathematical Sciences GC University, Lahore, Pakistan. DECLARATION I, Mr. Qaisar Latif Registration No. 105{GCU{PHD{SMS{09 student at Abdus Salam School of Mathematical Sciences GC University in the subject of Mathe- matics, hereby declare that the matter printed in this thesis, titled \Approximation properties and eigenvalue problems for Banach spaces" is my own work and that 1. No direct major work had already been done by me or anybody else on this topic; I worked on, for the PhD degree. 2. The work, I am submitting for the Ph. D. degree has not already been submitted elsewhere and shall not in future be submitted by me for obtaining similar degree from any other institution. Dated: |||||||||-, Signature of Deponent: |||||||||||| RESEARCH COMPLETION CERTIFICATE It is certified that the research work contained in this thesis, titled \Approximation properties and eigenvalue problems for Banach spaces" has been carried out and completed by Mr. Qaisar Latif, registration no. 105{GCU{ PHD{SMS{09 under my supervision. Date: |||||||{ Supervisor: |||||||||||| Dr. Oleg I. Reinov Submitted Through Prof. Dr. A. D. Raza Choudary |||||||||||{ Director General, Controller of Examination, Abdus Salam School of Mathematical Sciences, GC University, Lahore- GC University, Lahore, Pakistan Pakistan. To my family & friends Acknowledgement I would like to bring my deep acknowledgements to each and every person, institution and organisation for their efforts and supports in making Abdus Salam School of Mathematical Sciences (ASSMS) a success in mathematical sciences. There is no doubt that the efforts of Professor A. D. Raza Choudary for ASSMS and mathematics of Pakistan are incredibly nice and historical and, I believe, they will be followed as examples. He has always been encouraging to the students in both their difficult and good times and helping them discover themselves. Oleg Reinov, is indeed one of the top specialist in his field \Geometry of Banach spaces" currently and I am very lucky to have him as my PhD supervisor. He has provided me a sincere constant support in learning mathematics since my first semester at ASSMS. It was not really easy to work while I was away for Germany but he gave me a proper time through Skype and rapid email correspondence regardless of limited resources and time zone difference and hence this thesis is a result. His support is unforgettable. I am also very lucky to work with Alan Huckleberry at Jacobs university Bremen for my better understanding and study of complex analysis in the study of flag domains. He is accessible at any time and is always willing to discuss interesting mathematics or I should say he is willing to make mathematics interesting in a friendly discussion. He has really shaped my mathematics and taught me how to think in a right way and ask right questions. I would like to thank Jacobs university Bremen for providing me an excellent library and resources to help me improve my mathematics and complete this project. Many thanks for my family and friends who left me alone for work when there was a need and they provided me a company when needed. I am thankful for their understanding. Krankenhaus St. Josef Stift, Qaisar Latif. Bremen, Deutschland. November 21, 2013. i Abstract In the modern day language, finding quasi Banach operator ideals admitting eigenvalue type `s for 0 < s < 1 and those admitting some traces -for example the spectral trace, were two important problems set in 20th century. We set out with a description of eigenvalue type of Nr for 0 < r ≤ 1 restricted to subspaces of Lp-spaces and the coincidence of nuclear r;p and spectral traces on them. We proceed and consider operator ideals Nr;p and N -where p and r are related, and describe its eigenvalue type with many different techniques -all of which are important in themselves. We observe strong relationships among eigenvalue type, nuclear and spectral traces and approximation properties of type (r; p); APr;p for short. We prove that if 0 < s ≤ 1 and 1=r = 1=s + j1=p − 1=2j, then every Banach space r;p X has APr;p ( and also AP ). The optimality of our results for the eigenvalue types is provided to show that there is no hope for any improvements in it. A negative answer to a question of A. Pietsch and A. Hinrichs is provided; they asked for the validity of the d inclusion Nr ⊂ Nr, where 0 < r < 1. Independently and without knowing that it was considered by E. Oja et al., we have replied with a negative answer to the open question posed by two Indian mathematicians Anil K. Karn and D. P. Sinha. They didn't know if there existed a Banach space without approximation property of type p for 1 ≤ p < 2. Contents Introduction1 1 Foundations3 1.1 Quasi Banach spaces..............................3 1.2 Bounded Linear Operators...........................4 1.3 Open Mapping Theorem............................5 1.4 Quotient Spaces.................................5 1.5 Dual Space...................................6 1.6 Hahn-Banach Theorem.............................6 1.7 Bilinear Maps..................................7 1.8 Dual Of An Operator..............................7 1.9 Weak topology.................................8 1.10 Product of Banach Spaces...........................8 1.11 Some `p spaces.................................9 1.11.1 Weak `p spaces............................. 10 1.12 Tensor Product of Banach spaces....................... 11 1.12.1 Reasonable crossnorm......................... 12 1.12.2 Projective and injective tensor products............... 14 1.13 Operator Ideals................................. 14 1.13.1 Product of operator ideals....................... 15 1.13.2 Dual of an operator ideal........................ 16 1.13.3 Tensor stability of a quasi Banach operator ideal........... 16 1.14 Nuclear operator ideals............................. 17 1.15 Integral operator ideals............................. 18 1.16 Summing operator ideals............................ 18 1.17 Approximation Properties of type (r,p,q) and the like............ 19 1.18 Order boundedness............................... 20 1.19 Riesz Operators................................. 21 1.20 Related operators................................ 21 1.20.1 The principle of related operators................... 22 1.21 Eigenvalue type of an operator ideal...................... 22 1.21.1 Principle of uniform boundedness................... 22 1.22 Traces and determinants on an operator ideal................ 22 ii CONTENTS iii 1.23 Miscellaneous facts............................... 25 1.23.1 Lifting property of `1 .......................... 25 1.23.2 Hadamard Inequality.......................... 26 1.23.3 Two theorems of A. Pietsch...................... 26 1.23.4 Hadamard factorization type theorems................ 27 2 Grothendieck-Lidskiˇi trace formulas and some AP 's 28 2.1 Classical Grothendieck-Lidskiˇi Theorems................... 28 2.2 Grothendieck-Lidskiˇi Theorems........................ 29 2.2.1 Grothendieck-Lidskiˇi theorems for subspaces of Lp-spaces...... 29 2.2.2 \Helping" family of quasi Banach operator ideals Hp ........ 32 2.2.3 Grothendieck-Lidskiˇi theorems for subspaces of quotients of Lp-spaces 39 2.3 Applications of Grothendieck-Lidskiˇi Theorems............... 46 2.3.1 Using Fredholm theory......................... 48 r;p 3 Eigenvalue distribution of Nr;p, N and some AP 's 54 3.1 The eigenvalue distribution of Nr restricted to X(2;p) ............. 54 3.2 Nr;p restricted to X(2;p) and the eigenvalue type of operator Ideal Nr;p ... 57 3.3 Some Fredholm theory for Nr;p ........................ 58 3.3.1 Tensor stability and eigenvalue type of Nr;p ............. 60 3.4 General trace formulas and some AP 's.................... 62 3.4.1 Using Nr restricted to a class of Banach spaces........... 63 3.4.2 Using inclusion Nr;p ⊆ Nr0;p ...................... 65 3.4.3 Using Complex Analysis........................ 65 3.5 The eigenvalue type of operator ideal Nr;p .................. 67 3.5.1 Using Nr restricted to subspaces of quotients of Lp-spaces..... 67 3.5.2 Some Fredholm theory for Nr;p .................... 68 3.5.3 Tensor stability and the eigenvalue type of Nr;p ........... 71 3.6 General Trace formulas for Nr;p ........................ 74 3.7 A remark on APr ................................ 74 3.8 Counterexamples................................ 75 4 Banach spaces without AP of type p 83 5 An overview 88 Works presented in conferences 95 Bibliography 97 Introduction It was F. Riesz in 1918 who first proved that any compact operator admits countably many eigenvalues and moreover it converges to zero. But I. Schur described in 1909, the square summability of eigenvalues of an integral operator given by a continuous kernel. Therefore, it was set forth
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