Let $A$ be an operator acting on a Hilbert space. The (classical) {\it numerical range} $W(A)$ (also known as {\it the field of values}) of $A$ is the collection of the complex numbers of the form $(Ax,x)$ with $x$ ranging over all unit vectors in the Hilbert space. The radius of the smallest circle in the complex plane centered at the origin that encloses $W(A)$ is called the numerical radius $r(A)$ of $A$.

The study of the numerical range, numerical radius, and their generalizations has a long and distinguished history (e.g., see [GR] [HJ, Chapter 1]). Currently, there is a great deal of active research conducted by various research groups. A subject search on``numerical range'', ``numerical radius'' or ``field of values'' on the Math Sci Net will return over 500 items. Many Ph.D. theses have been devoted entirely or partly to this subject (e.g., those by N. Bebiano, C.A. Berger, C.F. Chan, C.M. Cheng, D.R. Farenick, I. Filippenko, C.K. Li, M.E. Lundquist, R.C. Mathias, T.Y. Tam, N.K. Tsing, J.L.M. van Dorsselaer, etc.). Also, there are monographs, special issues of journals, chapters of textbooks (e.g., see [AL, BD1, BD2, BD3, GR, Ha, HJ, I, SS]) devoted to the subject. Research papers on the subject continually appear in mathematical journals. Talks on the subject are frequently presented in mathematical conferences.

The subject is related and has applications to many different branches of pure and applied science such as operator theory, functional analysis, $C^*$-algebras, Banach algebras, matrix norms, inequalities, numerical analysis, perturbation theory, matrix polynomials, systems theory, quantum physics, etc. Moreover, a wide range of mathematical tools including algebra, analysis, geometry, combinatorial theory, and computer programming are useful in the study. These are some reasons why the subject attracts many researchers.

In the following, we describe some active research topics on the study of numerical ranges and numerical radii. To avoid a lengthy citation list, we do not include many references in our description. If desired, one can easily find the related references by doing keyword or author searches on the MathSciNet, or looking up the Mathematics Reviews.

** Quantum computing and quantum control **

One of the most exciting development is the study of rank-k numerical range of $A$ defined as the collection of complex number z for the existence of an n x k matrix X such that X*X = I_k and X*AX = zI_k. The definition is motivated by the study of quantum error correcting codes. One can go to goole.com and MathSciNet to search for higher rank numerical ranges to get the links to many papers on the topic. Also, the $C$-numerical range has been identified as a useful tool to study quantum control problems connected to NMR spectrocopy. Again, a goole search of the key word C-numerical range will lead to many relevant links.

** Operator theory and functional analysis **

As indicated in [BD1, BD2, BD3, GR, Ha, HJ, I], the numerical range and numerical radius are very useful for studying linear operators acting on Hilbert spaces or Banach spaces. Recently, the concepts have been usee to study linear or non-linear operators acting on more general spaces including $m$-convex algebras, topological $*$-algebras, $H$-locally convex spaces, Krein spaces, Wachs spaces, and inner product spaces over real quaternions.

There has also been interest in studying the numerical range of a special operator such as a (quasi)hyponormal operator, a Toeplitz operator with harmonic symbol, a tridiagonal operator, a weighted shift operator, a block shift operator, a dissipative operator, a quaternionic matrix, a doubly stochastic matrix, etc. In many instances, the numerical range can be used to classify special classes of operators (self-adjoint, normal, unitary), or characterize Banach $*$-algebras.

**Multilinear algebra**

For a given matrix or operator $A$, let $K(A)$ denote that induced operator of $A$ associated with a certain generalized matrix function. The {\it decomposable numerical range} of a matrix $A$ is the collection of complex numbers of the form $(K(A)x^*, x^*)$ with $x^*$ ranging through the decomposable unit tensors on the corresponding multilinear structure. Similar to the classical case, the decomposable numerical range is a useful tool for studying the induced operator $K(A)$, and results on the corresponding decomposable numerical radius always lead to interesting matrix inequalities. For example, one may see the work of N. Bebiano, C.F. Chan, C.M. Cheng, S. Hu, T.G. Lei, J. da Providencia, T.Y. Tam, F. Zhang, M. Marcus and his associates, etc.

** Several operator theory and system theory **

A generalization of the numerical range is the {\it joint numerical range} of a family of operators $(A_j)_{j \in J}$ for some index set $J$ defined as the collection of elements of the form $( (A_jx,x) )_{j \in J}$ with $x$ ranging over all unit vectors $x$. This concept is very useful in studying the joint behaviour of the operators $A_j$, and has applications to the joint spectrum, joint spectral norm, joint convergence of operators, structured singular values in system theory. A keyword search on joint numerical range on Math Sci Net will generate a long list of references.

** Location of spectrum and applications **

It is known that the classical numerical range of a matrix always contains its spectrum. As a result, the study of numerical range can help understand properties that depend on the location of the eigenvalues such as the stability and nonsingularity of matrices. Such a study has been done by many authors including W.V. Parker, F.L. Bauer, C.R. Johnson, and I. Spitkovsky.

Studies of the spectra of linear or non-linear operators on more general spaces using the numerical range have been done by R. Verma, Y.O. Yang, S. Young, S.J. Bhatt, etc.

In addition to the usual spectrum, the numerical range is useful in studying the perturbation of generalized eigenvalues of positive definite pencils (e.g., see [SS, Chapter VI] and its references).

** Quantum physics **

Many problems in quantum physics reduce to the study of expressions of the form ${\rm tr}(CU^*AU)$, where $U$ is unitary, for a fixed pair of matrices $A$ and $C$ (e.g., see [BR]). The collection of all such complex numbers is known as the $C$-{\it numerical range} of $A$, and has attracted the attention of many authors in the last few decades (e.g., see the survey article in [AL]). In particular, in the study of quantum dynamics, one would like to study the efficiency of unitary transformations between non-Hermitian states (e.g., see [N, St] and their references). It is equivalent to computing the $C$-numerical radius of $A$ for certain sparse nilpotent matrices $C$ and $A$.

The $(p,q)$ {\it numerical range} (alternatively, the {\it decomposable numerical range} of the $p$-{th} derivation of an operator over the $q$-{th} Grassmann space) studied by Bebiano, Li, Tam, Tsing, de Providencia, etc., is related to and is useful in the study of a $q$-particle system with $p$-body interaction in quantum physics (e.g., see [RS]).

** Norms **

A number of authors including Y. Nakamura, B. Mirman, E. Deutsch and C.M. McGregor have used the numerical range to study problems on norms of operators.

In the finite dimensional case, R.C. Mathias has shown that $C$-{\it numerical radii} (see the previous section for the definition of the $C$-numerical range) are the building blocks of {\it unitary similarity invariant norms} (also known as {\it weakly unitarily invariant norms}, e.g., see [AL] and [Bh] for the general background) on square matrices. Thus the study of $C$-numerical radii can help solve general problems involving unitary similarity invariant norms.

** Matrix inequalities **

As mentioned before, studying the decomposable numerical radius is useful in obtaining matrix inequalities. In addition, there has been a great deal of interest in studying inequalities relating various kinds of numerical radii and norms. Some of these questions are motivated by problems in applications.

** Numerical analysis **

Several authors have applied the theory of numerical range to study the convergence rate of various algorithms successfully. An excellent survey can be found in [GR, Chapter 4].

** Computer generation of numerical ranges **

In general, having an accurate plot of the numerical ranges would help one to get deeper insight about the theory of numerical ranges and numerical radii. There are several existing computer programs for plotting certain kinds of generalized numerical ranges (e.g., see [HJ], and http://www.math.wm.edu/$\,\tilde{}\,$ckli). However, how to improve the efficiency of the existing programs so that they can be used for practical problems, and how to design programs for generating other kinds of numerical ranges, especially those non-convex types, are interesting problems under current research.

** Other related subjects **

There are many other problems involving numerical ranges and numerical radius worth studying. For example, some authors have applied the theory of numerical ranges and numerical radii to study operator convergence properties, functional equations, operator trigonometry, model theory, robust stability, reduction theory, factorization of matrix polynomials, etc. In addition, there are many intrinsic problems such as the convexity of various kinds of generalized numerical ranges, the realizability of certain sets as the numerical ranges of an operator, the completability of partial matrices with some desired properties on the numerical ranges of the complete matrices, the classification of linear preservers of different types of numerical ranges and numerical radii, etc.

**
Including the following Proceedings of the workshops on
Numerical Ranges and Numerial radii.
**

[AL1] T. Ando and C.K. Li (special editors), Linear and Multilinear Algebra Vol. 37, nos. 1--3,1994.

[AL2] T. Ando and C.K. Li (special editors), Linear and Multilinear Algebra Vol.\ 43, no 4, 1998.

[LT1] C.K. Li and T.Y. Tam (special editors), Linear and Multilinear Algebra Vol.\ 52, nos 3--4, 2006.

[LT2] C.K. Li and T.Y. Tam (special editors), Linear and Multilinear Algebra Vol.\ 57, no 5 , 2009.

[LT2] C.K. Li, Y.T. Poon, and N.C. Wong (special editors), Linear and Multilinear Algebra Vol.\ 62, no 5 , 2014.

**
Additional references**

[Bh] R. Bhatia, Matrix Analysis, Springer-Verlag, New york, 1997.

[BD1] F.F. Bonsall and J. Duncan, Numerical Ranges, Vol. I, Cambridge University Press, 1971.

[BD2] F.F. Bonsall and J. Duncan, Numerical Ranges, Vol. II, Cambridge University Press, 1973.

[BD3] F.F. Bonsall and J. Duncan, Studies in Functional Analysis - Numerical Ranges, Studies in Mathematics Vol. 21, MAA, 1980.

[BR] J.-P. Blaizot and G. Ripka, Quantum Theory of Finite Systems, The MIT Press, Cambridge, Massachusetts, 1986.

[GR] K.E. Gustafson and D.K.M. Rao, Numerical Range: The Field of Values of Linear Operators and Matrices, Springer-Verlag, New york, 1997.

[I] B. Istratescu, Introduction to Linear Operator Theory, Marcel Dekker, New York, 1982.

[Ha] P.R. Halmos, A Hilbert Space Problem Book, Second Ed., Springer-Verlag, New York, 1982.

[HJ] R.A. Horn and C.R. Johnson, Topics in Matrix Analysis, Cambridge Univ. Press, Cambridge, 1991.

[N] N.C. Nielsen et.\ al., Bounds on spin dynamics tightened by permutation symmetry: Application to coherence transfer in $I_2S$ and $I_3S$ spin systems, Molecular Physics 85 (1995), 1205-1216.

[RS] P. Ring and P. Schuck, The Nuclear Many-Body Problem, Spring-Verlag, New York, 1980.

[SS] J.-g. Sun and G.W. Stewart, Matrix Perturbation Theory, Academic Press, New York, 1990.

[St] J. Stoustrup et.\ al., Generalized bound on quantum dynamics: Efficiency of unitary transforms between non-Hermitian states, Physical Review Letters 74 (1995), 2921-2924.

** Some reasons for meetings **

Since the study of numerical ranges and numerical radii is related to so many different branches of science, it is difficult to discuss the subject in a meeting of a special area such as functional analysis, numerical analysis, linear algebra, or operator theory. It is also difficult to discuss the subject in depth in a general mathematics conference. A more appropriate way to bring researchers on numerical ranges and numerical radii from different (research and geographic) areas together and provide a nice environment for them to exchange ideas is to have special meetings on the subject. Indeed, such workshops have taken place (see the next subsection for details.) In these workshops, interesting problems and ideas on the subjects have been exchanged, and important problems in applications have been identified. Some papers presented in the meetings were published in the research journal - \it Linear and Multilinear Algebra \rm - and have attracted the attention of other researchers, leading to further discovery of connections of the subject to other areas. It is clear that the past meetings have generated fruitful results. Here are some general goals of the meetings.

** Meetings **

In early seventies, numerical range workshops have been organized by Bonsall and Duncan. More meetings were organized starting from 1991, including the biennial workshop series on the subject. One may see