The Essentially Hankel Operators On Hardy Space

People take great interests in the Toeplitz operator and Hankel operator on H 2for their special construction and widespread application, over the decades years, many thoroughgoing and painstaking work had been done by some famous mathematicians. Going a step further, for these two calss of operators, we can get different kinds of generalizations of them.In the first part of this paper, we give a brief introduction of the two class of operators. We also introduced their recent research and generalizations in the following part. Gradually, after presenting some examples of essentially Hankel operator(a special generalization of Hankel operator),I focus my main attention on the proof of a relevant conclusion to the essentially Hankel operator.Here K is a compact operator, S is the unilateral forward shift, we define T is an essentially Hankel operator, when S * T-TS =K. For an operator which has the form "Hankel operator + compact operator" ,we can see obviously that it is Hankel indeed, then a natural problem arise: does all the essentially Hankel operator can be represent as the above form? The answer is negative, after presenting several typical examples (reference[1]) . In this paper, in a way of reduction to absurdity and construction, we prove the conclusion inextenso.