# Daoxing Xia

Professor Emeritus

#### Research Interests

Operator Theory and its Application, Functional Analysis, Applied Mathematics, Modern Mathematical Physics, Theory of Functions and Theory of Probability

#### Publications

### Books and Monographs

*Spectral Theory of Linear Operators*, Press Acad. Sinica, Beijing, (1987), 1-460.

*The Second of Functional Analysis*, Press High. Edu., Beijing, (1986). New edition (2009).

*An Invitation to the Theory of Linear Topological Spaces*, Sci. Tech. Press Shanghai (1986) 1-321.

*Spectral Theory of Hypononnal Operators*, Sci. Tech. Press, Shanghai, (1981), Birkhauser Verlag, Basel-Boston-Stuttgart, (1983) 1-241.

*Theory of a Real Variable and Functional Analysis*, Press Acad. Sinica, Beijing, (1980) 1-448.

*Measures and Integration on Infinite-dimensional spaces*, Sci. Tech. Press Shanghai, (1965), Acad. Press, N.Y., London (1972) 1-425. New edition, China High Education Press, Bejing (2009).

*Theory of Functions of a Real Variable and Essentials of Functional Analysis*, Sci. Tech. Press Shanghai (1956), 1-747. New edition, China Higher Education Press, Beijing (2010).

Analytic Theory of Subnormal Operator (in Chinese), China Higher Education Press, Beijing, Aug. 2012; (in English), World Scientific, New Jersey, London, Singapore, 2014.

### Articles

**83**(2015), 381-392.

**81**(2015), 237-254.

**77**(2013), 291-298.

"Analytic theory of hyponormal operators with rank one self-commutators", Integr. Equat. Oper. Theory,

**75**(2013), 573-589.

"Operator identities for subnormal tuples of operators," In "A Panorama of Modern Operator Theory and Related Topics," edited by H. Dym, M.A. Kaashoek, P. Lancaster, H. Langer, and L. Lerer in the series "Operator Theory: Advances and Applications," Vol. 218 (2012), 613-638, Birkhauser Verlag, Basel-Boston-Stuttgart.

"Determinant formula and trace formula for some finite rank perturbation of Heisenberg commutation relation," Integr. Equat. Oper. Theory, 60(2008), 573-590.

"Determinant formula for the trace class perturbation of Heisenberg commutation relation," Integr. Equat. Oper. Theory,

**56**(2006), 571-585.

**55**(2006), 439-452.

**54**(2006), 131-150.

*Integr. Equ. Oper. Theory*,

**48**(2004) 115-135.

*Integr. Equ. Oper. Theory*,

**47**(2003) 123-130.

*Integr. Equ. Opera Theory*,

**44**(2002) 370-382.

*Integr. Equ. Opera Theory*,

**38**(2001) 357-375.

*Integr. Eq. Opera Theory.*

**37**(2000), 487-504

*Integr. Equ. Opera Theory*

**33**(1999), 489-506.

"Chern characters associated with almost commuting algebras,"

*Proc. Amer. Math. Soc.*

**125**(1997), 507-514.

"Hyponormal operators with finite rank self-commutators and quadrature domains,"

*Jour. Math. Anal. Appl.*

**203**(1996), 540-559.

"On pure subnormal operators with finite rank self-commutators and related operator tuples,"

*Integ. Eq. Opera Theory*,

**24**(1996), 107-125. PDF

"Generalized cyclic cohomology associated with deformed commutators,"

*Proc. Amer.Math. Soc.*

**124**(1996), 1743-1753.

"Trace formulas and complete unitary invariants for some k-tuples of commuting operators,"

*Contem. Math.*

**185**(1995) 367-380.

"Trace formulas for the perturbation of some partial differential operators and cyclic cocycles on generalized Heisenberg group,"

*Opera Theory. Adv. and Appl.*

**62**, Birkhiiuser, Verlag, Basel (1993), 209-231.

"Trace formula for a class of subnormal tuple of operators,"

*Integr. Equ. Opera Theory.*

**17**(1993), 417-439.

"A note on the trace formulas for almost commuting operators,"

*Proc. Amer. Math. Soc.*

**16**(1992) 135-141.

"Complete unitary invariant for some subnormal operator,"

*Integr. Equ. Opera Theory.*

**15**(1992), 154-166.

*Integr. Equ. Opera Theory.*

**14**(1991), 390-399.

"Trace formulas for almost commuting operators, cyclic cohomology and subnormal operators,"

*Integr. Equ. Op. Theory.*

**14**(1991), 276-198.

"Some trace formulas for almost unperturbed SchrOdinger pairs of operators,"

*Proc. Amer. Math. Soc.*

**112**(1991), 755-764.

"Principal distribution for almost unpertubed Shrodinger pair of operators,"

*Proc. Amer. Math. Soc.*

**112**(1991), 745-754.

"On some classes of hypo normal tuples of commuting operators,"

*Oper. Theory: Advances and Applications*

**48**, Birkhiiuser, Verlag, Basel (1990), 423-448.

"Analytic theory of subnormal n-tuple of operators,"

*Proc. of Symp. in Pure Math.*

**51**(1990), part 1, 617-640.

"Cyclic cohomology and k-theory associated with difference operators,"

*Rocky Mtn. Jour. Math.*

**20**(1990), 637-650.

*Jour. Funct. Anal.*

**88**(1990), 1-63.

(with A. Aluthge) "A trace estimate of (

*T**

*T*)^

*P*-(

*TT**)^

*P*,"

*Integr. Equ. Oper. Theory.*

**12**(1989), 300-303.

"On the almost unperturbed Schrodinger pair of operators,"

*Integr. Equ. Oper. Theo.*

**12**(1989), 241-279.

"Cyclic cohomology associated with difference operators and anti-commutators,"

*Integr. Equ. Oper. Theory.*

**10**(1987),739-750.

"Trace formula for almost Lie algebra of operators and cyclic one cocyles,"

*Lecture Notes in Pure and Applied Mathematics*

**107**,

*Nonlinear and Convex Analysis*(1987) 299-308.

"Trace formula for almost Lie group of operators and cyclic one cocyles,"

*Integr. Equ. Oper. Theory.*

**9**(1986), 570-587.

"Trace formula for almost Lie algebra of operators,"

*Integr. Equ. Oper. Theory.*

**8**(1985), 854-881.

"Anti-commutators and the related trace formulas,"

*Integr. Equ. Oper. Theory.*

**8**(1985), 119-136.

*Integr. Equ. Oper. Theory.*

**8**(1984) 879-893.

**PDF 1 PDF 2**

"On locally bounded topological algebras,"

*Sci. Sinica*

**13**(1984), 375-390.

*Integr. Equ. Oper. Theory.*

**6**(1983), 507-514.

"On the kernels associated with a class of hypo normal operators,"

*Integr. Equ. Oper. Theory.*

**6**(1983), 444-452.

"On the analytic model of a class of hyponormal operators,"

*Integr. Equ. Oper. Theory.*

**6**(1983), 134-157.

"On the spectrum of hyponormal or semi-hyponormal operators,"

*J. Oper. Theory.*

**5**(1981), 257-266.

"Spectral mapping of hyponormal or semi-hyponormal operators,"

*Math. Anal. App.*

**79**(1981), 409-427.

(with S.Li) "On functional transformation of semi-hyponormal operators,"

*Chinese Ann. Math.*

**1**(1981), 501-504.

"On the Toeplitz operator related to the scattering operator of a semi-hyponormal operator,"

*Acta Math. Sci.*

**1**(1981), 77-82.

(with J. Pincus) "Mosaic and principal functions of hyponormal and semi-hyponormal operators,"

*Integr. Equ. Oper. Theory*

**4**(1981), 134-149.

"On the representation of the local current algebra,"

*Lecturer Notes on Math. Physics*, (1980), 311-315.

(with S.Li) "On the functional transformation of hyponormal operators,"

*Keuxue Tongbao*

**25**(1980), 625-627.

"On the scattering operator with indefmite metric,"

*Mathematical Problems in Theoretical Physics*, edited by K. Osterwalder, Springer-Verlag, 1980.

"On the representation of locally current algebras and group of diffeomorphisms,"

*Acta Sci. Sinica*, Special Issue II (1979), 149-160.

"On the commutational relation in the global gauge theory for group U(I),"

*Jour. Fudan Univ.*

**2**(1978),8-13.

(with S. Yan) "On field strengths, sources and gauge potentials of Yang-Mill's field,"

*Kexue Tongbao*

**22**(1977), 427-430,

*Jour. Fudan Univ.*

**1**(1977), 80-89.

"On field strengths and gauge potentials of Yang-Mill's fields,"

*Acta Sci. Sinica*

**20**(1977), 145-157.

"On the scattering problem concerning indefinite metric or with intermediate system,"

*Acta Math. Sinica*

**19**(1976), 39-51, 129-143.

"Abstract harmonic analysis on the quasi-invariant measures,"

*Collected Works on Mathematics of Fudan Univ*., (1966), 112-130.

"On completely non-unitary contractions,"

*Jour. Fudan Univ.*

**10**(1965).

"On some properties of quasi-invariant measures,"

*Acta Math. Sinica*

**6**(1965), 382-392.

"Positive-definite functions on commutative topological groups with quasi-invariant measures,"

*Acta Math. Sinica*

**5**(1964), 369-381,

*Acta Sci. Sinica*

**14**(1965), 653-665.

"On the quasi-invariant measures in linear topological spaces,"

*Jour. Fudan Univ.*

**83**(1963), 273-277.

"On the equivalence of a class of Gaussian measures,"

*Jour. Fudan Univ.*

**8**(1963), 151-161.

"On non-normal oprators (I),"

*Acta Math. Sinica*

**3**(1963), 232-246.

"On conditionally positive-defmite generalized functions,"

*Sci. Sinica*

**119**(1962), 1147-1168,

*Acta. Math. Sinica*

**113**(1961), 193-201.

(with L. Fan) "On the parametric representation of quasi-conformal mappings,"

*Sci. Sinica*

**11**(1962), 149-162.

"Measures on functional spaces,"

*Jour. Fudan Univ.*

**7**(1962), 121-131,

**8**(1965), 261-262,

*Acta Sci. Sinica*

**12**(1963), 479-494.

(with Z. Wu and W. Zhang) "On the semi-group of unitary operators on the indefmite metric space and the conditional positive-definite generalized functions,"

*Selected Works on Science in Shanghai*

**1**(1960), 28-45.

(with L. Fan) "On the parametric representation of quasi-conformal mappings and its applications,"

*Selected Works of Science in Shanghai*

**1**(1960), 66-99.

(with L. Fan) "On the modulus of quasi-conformal mapping,"

*Sci. Record, New Series*

**4**(1960),3 23-328.

(with S. Van) "On the distributions in the dual space of nuclear space,"

*Collected Works on Mathematics of Fudan Univ.*(1959), 413-418.

(with Z. Wu and W. Zhang) "On conditionally positive-definite generalized functions of several variables,"

*Collected Works on Mathematics of Fudan University*(1959), 437-448.

"On semi-normed rings with involution," (in Russian),

*Dokl. Akad.*

*Nauk, SSSR*

**124**(1959), 1223-1225,

*Izv. Adad. Nauk, SSSr. Ser. Mat.*

**23**(1959), 509-528.

"Positive functional on algebras," (in Russian),

*Dokl. Akad. Nauk, SSSR*

**124**(1959), 1223-1225, Izv

*. Akad. Nauk, SSS4, Ser. Mat.*

**23**(1959), 509-528.

"Parametric representation of quasi-conformal mappings,"

*Sci. Record, New Series*

**3**(1959), 400-407,

*Jour. Fudan Univ.*(1959), 26-44.

(with I.M. Gelfand) "On positively defmite generalized functions," (in Russian),

*Usp. Math. Nauk, SSSR*

**151**(1958).

"The method of variation and Loewner's method in the theory of Schlicht functions,"

*Jour. Fudan Univ.*

**1**(1958), 30-34.

"The minimum of the mean diameters of the Schlicht mappings,"

*Jour. Fudan Univ.*

**1**(1958), 28-29.

*Acta Math. Sinica*

**8**(1958), 408-412.

"Ring of analytic functions,"

*Jour. Furlan Univ.*

**2**(1957), 352-356.

"Typically-real functions on the multiply-connected domains,"

*Jour. Fudan Univ.*

**1**(1957), 138-145.

"Some covering properties of convex domains in the theory of conformal mapping,"

*Acta Math. Sinica*

**7**(1957), 421-432,

*Sci. Sinica*

**8**(1958), 816-828.

"On the radius of superiority in subordination,"

*Sci.*

*Record, New Series*

**1**(1957), 329-333.

"Goluzin's number (3-square root of 5)/2 is the radius of superiority in subordination,"

*Sci. Record*, New Series 1 (1957), 219-222.

"A shorter proof of Z. Charizynski's theorem,"

*Prog. Math*

**2**(1956), 714-720.

"An application of the theory of positive quadratic differential form to the product of mapping radii,"

*Acta Math. Sinica*

**6**(1956), 583-597.

"On the functions univalent in a circular ring,"

*Acta Math. Sinica*

**6**(1956), 598-618.

"Points exterior to the image domains for functions ofstarlikeness as well as of convexity,"

*Jour. Furlan Univ.*

**2**(1956), 125-132.

"On the dual quasi-invariant measures,"

*Acta Math. Sinica*

**6**(1956), 70-78.

The principle of area in the theory of univalent functions I,

*Jour. Furlan Univ.*

**2**(1956), 134-145.

"A covering theorem of a starlike domain,"

*Jour. Furlan Univ.*

**1**(1955), 51-54.

"On the moduli of some classes of analytic functions,"

*Acta Math. Sinica*

**5**(1955), 439-454.

"The products of mapping radii of non-overlapping domains,"

*Acta Math. Sinica*

**5**(1955), 27- 36.

"The principle of area in the theory of univalent functions,"

*Acta Math. Sinica*

**3**(1953), 208- 212.

"On the product of mapping radii for a system of non-overlapping domains,"

*Acta. Math. Sinica*

**3**(1953), 1-7.

"Theorems of distorsion and coefficients of Schlicht functions,"

*Sci. Record*

**4**(1951), 351-362.

"On the distorsion of Schlicht function,"

*Sci. Record*

**4**(1951), 209-212.

"On the coefficients of Schlicht functions,"

*Acta Math. Sinica*

**1**(1950), 98-107.

#### Conference Talks

2015: Conference on Functional Analysis, Shanghai, China

2014: International Conference on Analysis, Qongqing, China

2013: International Conference on Modern Analysis, Shanghai, China

2013: Chang Jianggong Research Institute of Mathematics, Hangzhou Normal University, Hangzhou, China

2012: Department of Mathematics, Chongqing University

2012: National Conference on Operator Theory, Chongqing, China

2011: SE Analysis Meeting, Giansville, FL

2009: International Conference on Operator Theory and Operator Algebra, Suzhou, China.

2008: Research Institute of Mathematics, Zhejiang University, China.

2008: Research Institute of Mathematics (Chern Institute), Nankai University, China.

2008: Department of Mathematics, Sandong University (Weihai Campus), China.

2008: Conference on Operator Theory (related to geometry), Quianhuandao, China.

2007: Department of Mathematics, Suzhou University, China.

2005: International Conference on Partial Differential Equations and Its Application to Geometry and Physics, Shanghai, China.

2005: SE Analysis Meeting, Lexington, VA.

2004: Special Session of American Mathematical Society, Nashville, TN

2004: Chinese Functional Analysis Conference, Shanghai.

2004: SE Analysis Meeting, Tuscaloosa, AL

2003: Analysis Seminar, Department of Mathematics, York University, Canada

2003: Seminar in Research Institute of Mathematics, and Colloquium in the Department of Mathematics, Fundan University, Shanghai

2002: Research Institute of Mathematics, Sangdong University, Jijiang, China

2001: Special Session, American Mathematical Society, New Orleans, LA

2000: Departments of mathematics, Zhejiang University, Nanjing University, China

1999: Department of Mathematics, Fudan University, Shanghai

1999: Special Session of American Mathematical Society Annual Meeting, Gainesville, FL

1998: SE Analysis Meeting, Tuscaloosa, AL (March)

1997: Analysis Seminar, Department of Mathematics, York University, CN

1997: SE Analysis Meeting, Gainesville, FL

1996: Special Session, American Mathematical Society, Iowa City, IA

1996: Research Institutes of Mathematics or Departments of Mathematics, Academy Sinica, Tapei and Beijing, National Central University, Taiwan, Normal University of East China, Shanghai, and National Jiao Tong University, Taiwan

1996: Annual Meeting of the American Mathematical Society, Orlando, FL

1995: Canadian Operator Symposium, Waterloo, CN

1995: SE Analysis Meeting, Atlanta, GA

1995: GPOTS, Cincinnati, OH

1995: Annual Meeting of the American Mathematical Society, San Francisco, CA

1995: International Workshop of Operator Theory & its Applications, Regensburg, Germany

1995: Canadian Operator Symposium, Waterloo, CN

1995: SE Conference on Analysis, Atlanta, GA

1995: GPOTS, Cincinnati, OH

1995: Annual Meeting of the American Mathematical Society, San Francisco, CA

1994: GPOTS, NE

1994: Seminar of Operator Theory, SUNY, Buffalo

1994: International Operator Theory Workshop, Banach Center, Polish Academy of Science

1994: SE Analysis Meeting, VA

1994: Summer Research Conference of Multivariable Operator Theory, Seattle, W A Functional Analysis Colloquium, University of California, Berkeley

1993: Departments of Mathematics, Hong Kong Chinese University, Hong Kong University of Science and Technology, Hong Kong University

1992: Special Session of American Mathematical Society, Tuscaloosa. AL

1992: SE Analysis Meeting, Knoxville, TN

1991: Seminar of Operator Theory, SUNY, Buffalo

1991: GPOTS, Austin, TX

1991: Conference of Analysis, Charlotte, NC

1991: Annual Meeting of the American Mathematical Society, San Francisco, CA

1990: Annual Meeting of the American Mathematical Society, Louisville, KY

1990: GPOTS, Albuquerque, NM

1989: Functional Analysis Seminars, SUNY at Buffalo and Stony Brook

1989: GPOTS, Houston, TX

1988: Functional Analysis Seminar, Purdue University

1988: Summer School of Operator Theory & Operator Algebra, New Hampshire

1988: SE Analysis Meeting, VA

1987: Analysis Meeting, Tuscaloosa. AL

1987: Great Plains Operator Theory Symposium (GPOTS), Lawrence, KS

1987: Conference of Functional Analysis, Tempe, AZ

1987: Departments of Mathematics, Purdue University, SUNY at Buffalo, and Indiana University Purdue University at Indianapolis

1987: Wabash Extramural Functional Analysis Seminar, IN

1986: Conference on Operator Theory, Bloomington, IN

1986: International Congress of Mathematics

1986: Southeast (SE) Analysis Meeting, Chapel Hill, NC SE

1985: Lectured in conferences in Berkeley, Santa Barbara, CA, Iowa City and NE

1984: Departments of Mathematics, University of Maryland, University of Florida, Wayne State University, Cleveland State University, Vanderbilt University, San Diego State University

1984: International Colloquium of Complex Analysis, Microlocal Calculus and Relativistic Quantum Theory, France

1983-1985: Fifth International Conference of Mathematical Physics, Switzerland

1983: Departments of Mathematics, University of Notre Dame, State University of Iowa, University of Delaware, University of Houston, Princeton

1982: Conferences of Hannonic Analysis, and Representations of Lie Groups, University Bielfield and Mathematisches Forschungsinstitiit Oberwolfach

1981: 6th International Conference of Mathematical Physics, West Berlin

1981: Department of Mathematics, Universitat Munchen

1980: Departments of Mathematics or Seminars of Functional Analysis and Operator Theory at the University of California, Berkeley, Los Angeles, Los Alamos Scientific Laboratory, Purdue University, Washington University, Yale University, University of Toronto, University of Pennsylvania, University of Maryland, Ohio State University, University of Michigan, Massachusetts Institute of Technology, University of Cincinnati, Kyushu University, and Nagoya University

1980: Two special sessions of American Mathematical Society, Colorado, and Pennsylvania

1979: Departments of Mathematics, Osaka University, University of Osaka Prefecture, Kyoto University and Tokyo University, Japan

#### Editorships

*Integral Equation and Operator Theory*, 1978-

*Operator Theory Advances and Applications*, 1978-

Editorial Board of

*Scientia Sinica*, 1972-1986

Editorial Board of

*Kexue Tongbao*(a monthly Journal of Science, Academia Sinica), 1972-1986

Deputy Editor of a series of monographs of

*Contemporary Fundamental Mathematics*(published by the press of the Academia Sinica) 1982-1990

Editorial Board of a series of monographs of

*Contemporary Mathematics*(published by the Shanghai Press of Science and Technology) 1982-1990

Deputy Editor of the

*Journal of Applied Sciences*, 1981-1982

Editor of the *Chinese Annals of Mathematics*, 1978-1996

#### Grants

6/1985 - 5/1987 NSF Grant #DMS-8502359

6/1987 - 5/1989 NSF Grant #DMS-8700048

6/1989 - 5/1991 NSF Grant #DMS-8901506

6/1991 - 5/1993 NSF Grant #DMS-9101268

6/1994 - 5/1997 NSF Grant #DMS-9400766

10/1998 - 2/1999 NSF Grant #DMS-9970813

10/2007 - 2/2008 NSF Grant #DMS-0800572

#### Awards

Third Place Chinese National Prize of Natural Sciences, 1982. For "Functional Integration and Spectral Theory of Operators" (with S.Z. Yan).

Fourth Place Chinese National Prize of Natural Sciences, 1982. For "Geometric Theory of Function" (with S.Gung).

#### Students

#### Ph.D. Students

Li, S.K., On the Non-nonnal Operators, 1982.

Tong, Y:S., Some New Results of the Spectral Analysis of Linear Operators on the Indefinite Inner Product Spaces, 1982.

Aluthge, A., Properties of

*p*-hyponormal operators, 1990.

Knisley, J.R., The analytic model of a semi-hyponormal operator with rank one polar difference, 1990.

Shirley, K.L., A scalar-valued analytic model for a class of subnormal operators, 1991.

Fryd; M, Taylor invertibility for commuting operator

*n*-tuples on Hilbert space, 1993.

Burton, D., On hyponormal tuples of commuting operators with fmite rank self-commutators, 1997.

Stewart, S.A., Some families of subnonnal operators with fmite ranak self-commutators, 2001.