In mathematics, particularly measure theory, the essential range of a function is intuitively the 'non-negligible' range of the function. One way of thinking of the essential range of a function is the set on which the range of the function is most 'concentrated'. The essential range can be defined for measurable real or complex-valued functions on a measure space.
... Th on Lp(μ) The operator norm of T is the essential supremum of h ... The essential range of h is defined in the following way a complex number λ is in the essential range of h if for all ε > 0, the preimage of the open ball Bε(λ) under h has strictly positive measure ... show first that σ(Th) coincides with the essential range of h and then examine its various parts ...
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