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Lionel Cooper | |
---|---|

JLB Cooper, c.1960 | |

Born | Beaufort West, Western Cape, South Africa | 27 December 1915

Died | 8 August 1979 63) London, England | (aged

Nationality | South African |

Alma mater | University of Cape Town Queen's College, Oxford |

Awards | Berwick Prize (1949) |

Scientific career | |

Fields | operator theory, transform theory, thermodynamics, functional analysis and differential equations |

Institutions | Birkbeck College Imperial College Cardiff University University of Witwatersrand Caltech University of Toronto. Chelsea College, University of London |

Doctoral advisor | Edward Titchmarsh |

Doctoral students | James Stewart |

**Jacob Lionel Bakst Cooper** (27 December 1915 – 8 August 1979) was a South African mathematician who worked in operator theory, transform theory, thermodynamics, functional analysis and differential equations.

**South Africa**, officially the **Republic of South Africa** (**RSA**), is the southernmost country in Africa. It is bounded to the south by 2,798 kilometres (1,739 mi) of coastline of Southern Africa stretching along the South Atlantic and Indian Oceans; to the north by the neighbouring countries of Namibia, Botswana, and Zimbabwe; and to the east and northeast by Mozambique and Eswatini (Swaziland); and it surrounds the enclaved country of Lesotho. South Africa is the largest country in Southern Africa and the 25th-largest country in the world by land area and, with over 57 million people, is the world's 24th-most populous nation. It is the southernmost country on the mainland of the Old World or the Eastern Hemisphere. About 80 percent of South Africans are of Sub-Saharan African ancestry, divided among a variety of ethnic groups speaking different African languages, nine of which have official status. The remaining population consists of Africa's largest communities of European (White), Asian (Indian), and multiracial (Coloured) ancestry.

In mathematics, **operator theory** is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators or closed operators, and consideration may be given to nonlinear operators. The study, which depends heavily on the topology of function spaces, is a branch of functional analysis.

In mathematics, **transform theory** is the study of transforms. The essence of transform theory is that by a suitable choice of basis for a vector space a problem may be simplified—or *diagonalized* as in spectral theory.

Cooper was born in Beaufort West, Western Cape, South Africa. Following the death of his father in 1919, the family moved to Cape Town where he attended the South African College School. His ability was soon recognized and he was encouraged to become a rabbi. However he rejected the Jewish faith and instead studied mathematics and physics at the University of Cape Town, where he won the Governor General’s prize for pure mathematics. He took an active part in student politics becoming a socialist member of the Student’s Parliament, with strong views against racism and Nazism.^{ [1] }

**Cape Town** is the oldest city in South Africa, colloquially named the Mother City. It is the legislative capital of South Africa and primate city of the Western Cape province. It forms part of the City of Cape Town metropolitan municipality.

The **South African College Schools** is a primary and secondary education institution in Newlands, Cape Town, South Africa. Founded in 1829, it is the oldest school in South Africa. SACS is one of four schools expressly named by Cecil John Rhodes to offer an annual Rhodes Scholarship to one of its graduating students. The schools are a combination of the South African College Junior School and the South African College High School.

In Judaism, a **rabbi** is a teacher of Torah. The basic form of the rabbi developed in the Pharisaic and Talmudic era, when learned teachers assembled to codify Judaism's written and oral laws. The first sage for whom the Mishnah uses the title of rabbi was Yohanan ben Zakkai, active in the early-to-mid first century CE. In more recent centuries, the duties of a rabbi became increasingly influenced by the duties of the Protestant Christian minister, hence the title "pulpit rabbis", and in 19th-century Germany and the United States rabbinic activities including sermons, pastoral counseling, and representing the community to the outside, all increased in importance.

He won a Rhodes Scholarship to Oxford and started to study at Queen's College in 1935,^{ [2] } obtaining his D.Phil. in 1940 under the supervision of Edward Charles Titchmarsh.^{ [3] } His work made an immediate impact and as early as January 1939 he was invited to speak at Hadamard’s seminar in the Collège de France.

The **Rhodes Scholarship** is an international postgraduate award for students to study at the University of Oxford. It was established in 1902, making it the first large-scale programme of international scholarship. The Rhodes Scholarship was founded by English businessman and politician Cecil John Rhodes, to promote unity between English speaking nations and instill a sense of civic-minded leadership and moral fortitude in future leaders irrespective of their chosen career paths. Although initially restricted to male applicants from countries which are today within the Commonwealth, as well as Germany and the United States, today the Scholarship is open to applicants from all backgrounds and from across the globe. Since its creation, controversy has surrounded both its former exclusion of women, and Rhodes' Anglo-supremacist beliefs and legacy of colonialism.

A **Doctor of Philosophy** is the highest university degree that is conferred after a course of study by universities in most English-speaking countries. PhDs are awarded for programs across the whole breadth of academic fields. As an earned research degree, those studying for a PhD are usually required to produce original research that expands the boundaries of knowledge, normally in the form of a thesis or dissertation, and defend their work against experts in the field. The completion of a PhD is often a requirement for employment as a university professor, researcher, or scientist in many fields. Individuals who have earned a Doctor of Philosophy degree may, in many jurisdictions, use the title *Doctor* or, in non-English-speaking countries, variants such as "Dr. phil." with their name, although the proper etiquette associated with this usage may also be subject to the professional ethics of their own scholarly field, culture, or society. Those who teach at universities or work in academic, educational, or research fields are usually addressed by this title "professionally and socially in a salutation or conversation." Alternatively, holders may use post-nominal letters such as "Ph.D.", "PhD", or "DPhil". It is, however, considered incorrect to use both the title and post-nominals at the same time.

**Edward Charles** "**Ted**" **Titchmarsh** was a leading English mathematician.

While in Oxford he joined the Communist party, where he met his future wife Kathleen Dixon. Turned down for active service due to his poor eyesight, he worked for the Bristol Aeroplane Company from 1940 to 1944. During this time he became disenchanted with Russia’s non-aggression pact with the Nazis and left the Communist party.

The **Bristol Aeroplane Company**, originally the British and Colonial Aeroplane Company, was both one of the first and one of the most important British aviation companies, designing and manufacturing both airframes and aircraft engines. Notable aircraft produced by the company include the 'Boxkite', the Bristol Fighter, the Bulldog, the Blenheim, the Beaufighter, and the Britannia, and much of the preliminary work which led to the Concorde was carried out by the company. In 1956 its major operations were split into **Bristol Aircraft** and **Bristol Aero Engines**. In 1959, Bristol Aircraft merged with several major British aircraft companies to form the British Aircraft Corporation (BAC) and Bristol Aero Engines merged with Armstrong Siddeley to form Bristol Siddeley.

**Russia**, officially the **Russian Federation**, is a transcontinental country in Eastern Europe and North Asia. At 17,125,200 square kilometres (6,612,100 sq mi), Russia is by a considerable margin the largest country in the world by area, covering more than one-eighth of the Earth's inhabited land area, and the ninth most populous, with about 146.77 million people as of 2019, including Crimea. About 77% of the population live in the western, European part of the country. Russia's capital, Moscow, is one of the largest cities in the world and the second largest city in Europe; other major cities include Saint Petersburg, Novosibirsk, Yekaterinburg and Nizhny Novgorod. Extending across the entirety of Northern Asia and much of Eastern Europe, Russia spans eleven time zones and incorporates a wide range of environments and landforms. From northwest to southeast, Russia shares land borders with Norway, Finland, Estonia, Latvia, Lithuania and Poland, Belarus, Ukraine, Georgia, Azerbaijan, Kazakhstan, China, Mongolia and North Korea. It shares maritime borders with Japan by the Sea of Okhotsk and the U.S. state of Alaska across the Bering Strait. However, Russia recognises two more countries that border it, Abkhazia and South Ossetia, both of which are internationally recognized as parts of Georgia.

From 1944 he lectured at Birkbeck College and for a short time Imperial College, and was awarded the Berwick Prize in 1949. His correspondence of that year included two letters from Albert Einstein concerning possible logical inconsistencies in quantum mechanics.

**Birkbeck, University of London**, is a public research university located in Bloomsbury London, England, and a constituent college of the federal University of London. Established in 1823 as the **London Mechanics' Institute** by its founder, Sir George Birkbeck, and its supporters, Jeremy Bentham, J. C. Hobhouse and Henry Brougham, Birkbeck has been one of the few institutions to specialise in evening higher education.

**Albert Einstein** was a German-born theoretical physicist who developed the theory of relativity, one of the two pillars of modern physics. His work is also known for its influence on the philosophy of science. He is best known to the general public for his mass–energy equivalence formula *E* = *mc*^{2}, which has been dubbed "the world's most famous equation". He received the 1921 Nobel Prize in Physics "for his services to theoretical physics, and especially for his discovery of the law of the photoelectric effect", a pivotal step in the development of quantum theory.

**Quantum mechanics**, including quantum field theory, is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles.

In 1951 he was appointed professor at the Cardiff University ^{ [2] } where he stayed until 1963 apart from a break in 1954 when he spent some time in the University of Witwatersrand. From 1952 to 1959 he edited the proceedings of the London Mathematical Society. After leaving Cardiff he spent three years in North America, as visiting professor at Caltech during 1964, followed by a two-year stay at the University of Toronto. While in Toronto he edited the Canadian Journal of Mathematics.

**Cardiff University** is a public research university in Cardiff, Wales. Founded in 1883 as the **University College of South Wales and Monmouthshire**, it became one of the founding colleges of the University of Wales in 1893, and in 1997 received its own degree-awarding powers. It merged with the University of Wales Institute of Science and Technology (UWIST) in 1988. The college adopted the public name of Cardiff University in 1999, and in 2005 this became its legal name, when it became an independent university awarding its own degrees. The third oldest university institution in Wales, it is composed of three colleges: Arts, Humanities and Social Sciences; Biomedical and Life Sciences; and Physical Sciences and Engineering.

**The London Mathematical Society** (**LMS**) is one of the United Kingdom's learned societies for mathematics.

The **California Institute of Technology **(**Caltech**) is a private doctorate-granting research university in Pasadena, California. Known for its strength in natural science and engineering, Caltech is often ranked as one of the world's top-ten universities.

In 1967 he returned to the UK to serve as head of the mathematics department at the newly constituted Chelsea College, University of London, where he remained until his death.

Within operator theory, Cooper worked in the area of linear operators on real or complex Hilbert spaces. He studied the unbounded operators that arose from quantum theory,^{ [2] } extending basic work of Frigyes Riesz and John von Neumann.

Within transform theory, he worked on the representation and uniqueness of integral transforms, on approximation, and on linear transformations that satisfy functional relations arising from representations of linear groups.^{ [2] } In this he collaborated closely with P.L. Butzer of RWTH Aachen.

**Alain Connes** is a French mathematician, currently Professor at the Collège de France, IHÉS, Ohio State University and Vanderbilt University. He was an Invited Professor at the Conservatoire national des arts et métiers (2000).

In quantum mechanics, **bra–ket notation** is a standard notation for describing quantum states. It can also be used to denote abstract vectors and linear functionals in mathematics. The notation uses angle brackets and a vertical bar, to denote the scalar product of vectors or the action of a linear functional on a vector in a complex vector space. The scalar product or action is written as

**Functional analysis** is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations.

The **mathematical formulations of quantum mechanics** are those mathematical formalisms that permit a rigorous description of quantum mechanics. Such are distinguished from mathematical formalisms for theories developed prior to the early 1900s by the use of abstract mathematical structures, such as infinite-dimensional Hilbert spaces and operators on these spaces. Many of these structures are drawn from functional analysis, a research area within pure mathematics that was influenced in part by the needs of quantum mechanics. In brief, values of physical observables such as energy and momentum were no longer considered as values of functions on phase space, but as eigenvalues; more precisely as spectral values of linear operators in Hilbert space.

In mathematics, **spectral theory** is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result of studies of linear algebra and the solutions of systems of linear equations and their generalizations. The theory is connected to that of analytic functions because the spectral properties of an operator are related to analytic functions of the spectral parameter.

**Marshall Harvey Stone** was an American mathematician who contributed to real analysis, functional analysis, topology and the study of Boolean algebras.

**Mark Aronovich Naimark** was a Soviet mathematician who made important contributions to functional analysis and mathematical physics.

In mathematics, **generalized functions**, or **distributions**, are objects extending the notion of functions. There is more than one recognized theory. Generalized functions are especially useful in making discontinuous functions more like smooth functions, and describing discrete physical phenomena such as point charges. They are applied extensively, especially in physics and engineering.

**Jacob Theodore "Jack" Schwartz** was an American mathematician, computer scientist, and professor of computer science at the New York University Courant Institute of Mathematical Sciences. He was the designer of the SETL programming language and started the NYU Ultracomputer project. He founded the New York University Department of Computer Science, chairing it from 1964 to 1980.

**Irving Ezra Segal** was an American mathematician known for work on theoretical quantum mechanics. He shares credit for what is often referred to as the Segal–Shale–Weil representation.

**Franz Rellich** was an Austrian-German mathematician. He made important contributions in mathematical physics, in particular for the foundations of quantum mechanics and for the theory of partial differential equations. The Rellich–Kondrachov theorem is named after him.

**Jonathan R. Partington** is an English mathematician who is Professor of Pure Mathematics at the University of Leeds.

**Gerard J. Murphy** MRIA was a prolific Irish mathematician. His textbooks are internationally acclaimed, and translated into different languages. He died from cancer in October 2006, at the age of 57.

**Daoxing Xia** is a Chinese American mathematician. He is currently a professor at the Department of Mathematics, Vanderbilt University in the United States. He was elected an academician of the Chinese Academy of Science in 1980.

**Israel Gohberg** was a Bessarabian-born Soviet and Israeli mathematician, most known for his work in operator theory and functional analysis, in particular linear operators and integral equations.

**Nelson James Dunford** was an American mathematician, known for his work in functional analysis, namely integration of vector valued functions, ergodic theory, and linear operators. The Dunford decomposition, Dunford–Pettis property, and Dunford-Schwartz theorem bear his name.

**Walter Warwick Sawyer** (1911–2008) was a mathematician, mathematics educator and author, who taught on several continents.

**John P. Perdew** is a theoretical condensed matter physicist known for his contributions to the fields of solid-state physics and quantum chemistry. His work on density functional theory has led to him being one of the world's most cited physicists. Perdew currently teaches and conducts research at Temple University.

**Tosio Kato** was a Japanese mathematician who worked with partial differential equations, mathematical physics and functional analysis.

- ↑ Edmunds D.E:
*Jacob Lionel Bakst Cooper*, Bull London Math. Soc., 13 1981 pp 429-450 - 1 2 3 4 O'Connor, J. J.; Robertson, E. F. (December 1996). "Jacob Lionel Bakst Cooper". MacTutor. Retrieved 2008-10-15.
- ↑ "Jacob Lionel Bakst Cooper". Mathematics Genealogy Project. Retrieved 2008-10-15.

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