Since oblique rotation means that your factors are already correlated, finding cross-loadings indicates that the item(s) in question do not discriminate between those two factors. With Exploratory Factor Analysis, the tradition has been to eliminate that variable so that the solution exhibits "simple structure" with each variable loading on one and only factor, but that may not be the best solution.

An analogy would be to run a Confirmatory Factor Analysis with and without this cross-loading. The model without would show a notable "modification index" for the cross-loading and model with it would be a better fit. In that case, the usual choice would be to accept the better fitting but more complex model.

Thank you for your answer, prof. Morgan. Actually, I did not apply EFA, but item analysis (based on classical test theory) to test predicted item clusters (as an alternative to CFA). In the output of item analysis, two correlating clusters will show several cross-correlations between the items that are part of both. I have devised a goodness-of-fit measure, not based on a residual matrix as in CFA and exploratory structural equation modeling (ESEM), but on the correspondence between predicted and empirically found item clusters (or factors as defined by their indicators). This alternative measure can be affected unfavorably by cross-loading items, even though both the cluster (factor) correlations and cross-loading of the items had been anticipated and are actually confirming one’s model.

I suppose that in EFA with orthogonal rotation such items will be the ones that are clearly cross-loading on the factors corresponding with these clusters. I wonder: if one runs an oblique rotation, will these cross-loadings be much reduced as a result of allowing that factors to be correlated? If so, then my GOF-measure would no longer be affected unfavorably by such items, and it would be better to use ESEM instead of item analysis in order to find the empirical counterparts of one’s predicted factors. If not, perhaps one should use the β-coefficients of the factor pattern instead of the loadings in the factor structure to apply this GOF-measure on.

I do not have the equipment to apply EFA or ESEM in order to find out experimentally, hence my question.

Unless you have a strong reason for believing that your scales are indeed uncorrelated, I would recommend allowing them to be correlated in CFA (or equivalently an oblique rotation in EFA). And if you are using CFA, you can examine the Goodness of Fit measures for models with and without those correlations.

I don't know if you did the following, but it is quite common to run orthogonal rotations, then create scales by summing rather than using factor scores, and which can produce substantial correlations among those scales.

In general, ask yourself this: What names did you give your factors and would you truly expect measures of those concepts to be uncorrelated?

I have a question respected sir

My model fit is coming good with respect to CMIN/DF, CFI, NFI, RMSEA

Convergent validity also met but, problem with discriminant validity where, the value of MSV coming more as compared to AVE,

what should be the solution for this?

kindly guide

Raiswa, I advise you to ask your question to the RG participants in general. Add more information about your research subject, measurement instrument(s), model, and fit-indices inspected. Clarify the less common abbreviations such as MSV and AVE. Report also chi-square, its df, and its significance value. And how you determined the instrument's discriminant validity. People more acquainted with structural equation modeling than I am, will then be in a position to answer your question. As it is presented now, nobody will be able to answer your question.