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Toeplitz Operator
In operator theory, a Toeplitz operator is the compression of a multiplication operator on the circle to the Hardy space. Details Let ''S''1 be the circle, with the standard Lebesgue measure, and ''L''2(''S''1) be the Hilbert space of square-integrable functions. A bounded measurable function ''g'' on ''S''1 defines a multiplication operator ''Mg'' on ''L''2(''S''1). Let ''P'' be the projection from ''L''2(''S''1) onto the Hardy space ''H''2. The ''Toeplitz operator with symbol g'' is defined by :T_g = P M_g \vert_, where " , " means restriction. A bounded operator on ''H''2 is Toeplitz if and only if its matrix representation, in the basis , has constant diagonals. Theorems * Theorem: If g is continuous, then T_g - \lambda is Fredholm if and only if \lambda is not in the set g(S^1). If it is Fredholm, its index is minus the winding number of the curve traced out by g with respect to the origin. For a proof, see . He attributes the theorem to Mark Krein, Harold Widom, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operator Theory
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. If a collection of operators forms an algebra over a field, then it is an operator algebra. The description of operator algebras is part of operator theory. Single operator theory Single operator theory deals with the properties and classification of operators, considered one at a time. For example, the classification of normal operators in terms of their spectra falls into this category. Spectrum of operators The spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides cond ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operator Theory
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. If a collection of operators forms an algebra over a field, then it is an operator algebra. The description of operator algebras is part of operator theory. Single operator theory Single operator theory deals with the properties and classification of operators, considered one at a time. For example, the classification of normal operators in terms of their spectra falls into this category. Spectrum of operators The spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides cond ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Birkhäuser
Birkhäuser was a Swiss publisher founded in 1879 by Emil Birkhäuser. It was acquired by Springer Science+Business Media in 1985. Today it is an imprint used by two companies in unrelated fields: * Springer continues to publish science (particularly: history of science, geosciences, computer science) and mathematics books and journals under the Birkhäuser imprint (with a leaf logo) sometimes called Birkhäuser Science. * Birkhäuser Verlag – an architecture and design publishing company was (re)created in 2010 when Springer sold its design and architecture segment to ACTAR. The resulting Spanish-Swiss company was then called ActarBirkhäuser. After a bankruptcy, in 2012 Birkhäuser Verlag was sold again, this time to De Gruyter. Additionally, the Reinach-based printer Birkhäuser+GBC operates independently of the above, being now owned by ''Basler Zeitung''. History The original Swiss publishers program focused on regional literature. In the 1920s the sons of Emil Birkh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Operator
In functional analysis, a branch of mathematics, a compact operator is a linear operator T: X \to Y, where X,Y are normed vector spaces, with the property that T maps bounded subsets of X to relatively compact subsets of Y (subsets with compact closure in Y). Such an operator is necessarily a bounded operator, and so continuous. Some authors require that X,Y are Banach, but the definition can be extended to more general spaces. Any bounded operator ''T'' that has finite rank is a compact operator; indeed, the class of compact operators is a natural generalization of the class of finite-rank operators in an infinite-dimensional setting. When ''Y'' is a Hilbert space, it is true that any compact operator is a limit of finite-rank operators, so that the class of compact operators can be defined alternatively as the closure of the set of finite-rank operators in the norm topology. Whether this was true in general for Banach spaces (the approximation property) was an unsolved question ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Donald Sarason
Donald Erik Sarason (January 26, 1933 – April 8, 2017) was an American mathematician who made fundamental advances in the areas of Hardy space theory and VMO. He was one of the most popular doctoral advisors in the Mathematics Department at UC Berkeley. He supervised 39 Ph.D. theses at UC Berkeley. Education *B.S. in Physics from the University of Michigan in 1955. *Master's degree (A.M.) in Physics from the University of Michigan in 1957. *Ph.D. in Mathematics from the University of Michigan in 1963. Doctoral thesis supervised by Paul Halmos. Career Postdoc at the Institute for Advanced Study in 1963–1964, supported by a National Science Foundation Postdoctoral Fellowship. Then Sarason went to the University of California Berkeley as an Assistant Professor (1964–1967), Associate Professor (1967–1970) and until his retirement, Professor (1970–2012). Accomplishments Sarason was awarded a Sloan Fellowship for 1969–1971. Sarason was the author of 78 mathematics publ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sun-Yung Alice Chang
Sun-Yung Alice Chang (, hak, Chông Sṳn-yùng, ; born 1948) is a Taiwanese American mathematician specializing in aspects of mathematical analysis ranging from harmonic analysis and partial differential equations to differential geometry. She is the Eugene Higgins Professor of Mathematics at Princeton University. Life Chang was born in Xian, China in 1948 and grew up in Taiwan. She received her Bachelor of Science degree in 1970 from National Taiwan University, and her doctorate in 1974 from the University of California, Berkeley. At Berkeley, Chang wrote her thesis on the study of bounded analytic functions. Chang became a full professor at UCLA in 1980 before moving to Princeton in 1998. Career and research Chang's research interests include the study of geometric types of nonlinear partial differential equations and problems in isospectral geometry. Working with her husband Paul Yang and others, she produced contributions to differential equations in relation to geome ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sheldon Axler
Sheldon Jay Axler (born November 6, 1949, Philadelphia) is an American mathematician and textbook author. He is a professor of mathematics and the Dean of the College of Science and Engineering at San Francisco State University. He graduated from Miami Palmetto Senior High School in Miami, Florida in 1967. He obtained his AB in mathematics with highest honors at Princeton University (1971) and his PhD in mathematics, under professor Donald Sarason, from the University of California, Berkeley, with the dissertation "Subalgebras of L^" in 1975. As a postdoc, he was a C. L. E. Moore instructor at the Massachusetts Institute of Technology. He taught for many years and became a full professor at Michigan State University. In 1997, Axler moved to San Francisco State University, where he became the chair of the Mathematics Department. Axler received the Lester R. Ford Award for expository writing in 1996 from the Mathematical Association of America for a paper titled "Down with D ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harold Widom
Harold Widom (September 23, 1932 – January 20, 2021) was an American mathematician best known for his contributions to operator theory and random matrices. He was appointed to the Department of Mathematics at the University of California, Santa Cruz in 1968 and became professor emeritus in 1994. Education and research Widom was born in Newark, New Jersey. He studied at Stuyvesant High School, graduating in 1949, and was a member of the school's math team along with his brother Benjamin Widom (1944, 1948). Widom attended City College of New York until 1951, during which he was one of the winners of the William Lowell Putnam Mathematical Competition (1951). At the University of Chicago he obtained an M.S. (1952) and Ph.D., the latter on a thesis ''Embedding of AW*-algebras'' advised by Irving Kaplansky (1955). He taught mathematics at Cornell University (1955–68) where he started his work on Toeplitz and Wiener-Hopf operators, partly inspired by Mark Kac. Widom was appoi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dilation (operator Theory)
In operator theory, a dilation of an operator ''T'' on a Hilbert space ''H'' is an operator on a larger Hilbert space ''K'', whose restriction to ''H'' composed with the orthogonal projection onto ''H'' is ''T''. More formally, let ''T'' be a bounded operator on some Hilbert space ''H'', and ''H'' be a subspace of a larger Hilbert space '' H' ''. A bounded operator ''V'' on '' H' '' is a dilation of T if :P_H \; V , _H = T where P_H is an orthogonal projection on ''H''. ''V'' is said to be a unitary dilation (respectively, normal, isometric, etc.) if ''V'' is unitary (respectively, normal, isometric, etc.). ''T'' is said to be a compression of ''V''. If an operator ''T'' has a spectral set X, we say that ''V'' is a normal boundary dilation or a normal \partial X dilation if ''V'' is a normal dilation of ''T'' and \sigma(V)\subseteq \partial X. Some texts impose an additional condition. Namely, that a dilation satisfy the following (calculus) property: :P_H \; f(V) , _H = f(T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mark Krein
Mark Grigorievich Krein ( uk, Марко́ Григо́рович Крейн, russian: Марк Григо́рьевич Крейн; 3 April 1907 – 17 October 1989) was a Soviet mathematician, one of the major figures of the Soviet school of functional analysis. He is known for works in operator theory (in close connection with concrete problems coming from mathematical physics), the problem of moments, classical analysis and representation theory. He was born in Kyiv, leaving home at age 17 to go to Odessa. He had a difficult academic career, not completing his first degree and constantly being troubled by anti-Semitic discrimination. His supervisor was Nikolai Chebotaryov. He was awarded the Wolf Prize in Mathematics in 1982 (jointly with Hassler Whitney), but was not allowed to attend the ceremony. David Milman, Mark Naimark, Israel Gohberg, Vadym Adamyan, Mikhail Livsic and other known mathematicians were his students. He died in Odessa. On 14 January 2008, the memo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fredholm Operator
In mathematics, Fredholm operators are certain operators that arise in the Fredholm theory of integral equations. They are named in honour of Erik Ivar Fredholm. By definition, a Fredholm operator is a bounded linear operator ''T'' : ''X'' → ''Y'' between two Banach spaces with finite-dimensional kernel \ker T and finite-dimensional (algebraic) cokernel \mathrm\,T = Y/\mathrm\,T, and with closed range \mathrm\,T. The last condition is actually redundant. The '' index'' of a Fredholm operator is the integer : \mathrm\,T := \dim \ker T - \mathrm\,\mathrm\,T or in other words, : \mathrm\,T := \dim \ker T - \mathrm\,\mathrm\,T. Properties Intuitively, Fredholm operators are those operators that are invertible "if finite-dimensional effects are ignored." The formally correct statement follows. A bounded operator ''T'' : ''X'' → ''Y'' between Banach spaces ''X'' and ''Y'' is Fredholm if and only if it is invertible modulo compact ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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