Product numerical range
Given a Hilbert space with a tensor product structure a product numerical range is defined as a numerical range with respect to the subset of product vectors. In some situations, especially in the context of quantum mechanics product numerical range is known as local numerical range Introduction. Let formula_1 be an operator acting on an formula_2-dimensional Hilbert space formula_3. Let formula_4 denote its "numerical range", "i.e." the set of all formula_5 such that there exists a normalized state formula_6 where formula_7 and formula_8 are normalized. Product numerical radius. Let formula_9 be a tensor product Hilbert space . We define the "product numerical radius" formula_10 of formula_1, with respect to this tensor product structure, as formula_12 Notation. The notion of numerical range of a given operator, also called "field of values", has been extensively studied during the last few decades and its usefulness in quantum theory has been emphasized. Several generalizations of numerical range are known. In particular, Marcus introduced the notion of ’’’decomposable numerical range’’’, the properties of which are a subject of considerable interest. The product numerical range can be considered as a particular case of the decomposable numerical range defined for operators acting on a  tensor product Hilbert space. This notion may also be considered as a numerical range "relative" to the proper subgroup formula_13 of the full unitary group formula_14. General case. It is not difficult to establish the basic properties of the product numerical range which are independent of the partition of the Hilbert space and of the structure of the operator. We list them below leaving some simple items without a proof. Basic properties. Topological facts concerning product numerical range for general operators. Convexity. The product numerical range does not need to be convex. Consider the following simple example. Let Matrix formula_39  defined above is matrix with eigenvalues formula_40. It is easy to see that formula_41 and formula_42, but formula_43. Actually, by direct computation we have formula_44 Product numerical range of matrix formula_39 is presented below. Product numerical range forms a nonempty set for a general operator. In particular it contains the barycenter of the spectrum. Barycenter. Product numerical range of formula_46 includes the barycenter of the spectrum, formula_47 Product numerical radius is a vector norm on matrices, but it is not a matrix norm. Product numerical radius is invariant with respect to local unitaries, which have the tensor product structure. Product